The document summarizes a talk on obtaining subexponential time algorithms for NP-hard problems on planar graphs. It discusses using treewidth and tree decompositions to solve problems like 3-coloring in 2O(√n) time on n-vertex planar graphs. It also discusses the exponential time hypothesis and how it implies lower bounds, showing these algorithms are optimal up to constant factors in the exponent. The document outlines several chapters, including using grid minors and bidimensionality to obtain 2O(√k) algorithms for problems like k-path, even for some W[1]-hard problems parameterized by k.
Some fixed point theorems of expansion mapping in g-metric spacesinventionjournals
Over the past two decades the development of fixed point theory in metric spaces has attracted
considerable attention due to numerous applications in areas such as variation and linear inequalities,
optimization and approximation theory. Therefore, different Authors proved many fixed points results for self
mapping defined on complete G-Metric space. The objectives of this study are to prove fixed point results for
mapping satisfying expansion conditions.
We continue the study of the concepts of minimality and homogeneity in the fuzzy context. Concretely, we introduce two new notions of minimality in fuzzy bitopological spaces which are called minimal fuzzy open set and pairwise minimal fuzzy open set. Several relationships between such notions and a known one are given. Also, we provide results about the transformation of minimal, and pairwise minimal fuzzy open sets of a fuzzy bitopological space, via fuzzy continuous and fuzzy open mappings, and pairwise continuous and pairwise open mappings, respectively. Moreover, we present two new notions of homogeneity in the fuzzy framework. We introduce the notions of homogeneous and pairwise homogeneous fuzzy bitopological spaces. Several relationships between such notions and a known one are given. And, some connections between minimality and homogeneity are given. Finally, we show that cut bitopological spaces of a homogeneous (resp. pairwise homogeneous) fuzzy bitopological space are homogeneous (resp. pairwise homogeneous) but not conversely.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Some fixed point theorems of expansion mapping in g-metric spacesinventionjournals
Over the past two decades the development of fixed point theory in metric spaces has attracted
considerable attention due to numerous applications in areas such as variation and linear inequalities,
optimization and approximation theory. Therefore, different Authors proved many fixed points results for self
mapping defined on complete G-Metric space. The objectives of this study are to prove fixed point results for
mapping satisfying expansion conditions.
We continue the study of the concepts of minimality and homogeneity in the fuzzy context. Concretely, we introduce two new notions of minimality in fuzzy bitopological spaces which are called minimal fuzzy open set and pairwise minimal fuzzy open set. Several relationships between such notions and a known one are given. Also, we provide results about the transformation of minimal, and pairwise minimal fuzzy open sets of a fuzzy bitopological space, via fuzzy continuous and fuzzy open mappings, and pairwise continuous and pairwise open mappings, respectively. Moreover, we present two new notions of homogeneity in the fuzzy framework. We introduce the notions of homogeneous and pairwise homogeneous fuzzy bitopological spaces. Several relationships between such notions and a known one are given. And, some connections between minimality and homogeneity are given. Finally, we show that cut bitopological spaces of a homogeneous (resp. pairwise homogeneous) fuzzy bitopological space are homogeneous (resp. pairwise homogeneous) but not conversely.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Tree and Binary search tree in data structure.
The complete explanation of working of trees and Binary Search Tree is given. It is discussed such a way that everyone can easily understand it. Trees have great role in the data structures.
Trees. Defining, Creating and Traversing Trees. Traversing the File System
Binary Search Trees. Balanced Trees
Graphs and Graphs Traversal Algorithms
Exercises: Working with Trees and Graphs
this presentation is made for the students who finds data structures a complex subject
this will help students to grab the various topics of data structures with simple presentation techniques
best regards
BCA group
(pooja,shaifali,richa,trishla,rani,pallavi,shivani)
In many scientific areas, systems can be described as interaction networks where elements correspond to vertices and interactions to edges. A variety of problems in those fields can deal with network comparison and characterization.
The problem of comparing and characterizing networks is the task of measuring their structural similarity and finding characteristics which capture structural information. In order to analyze complex networks, several methods can be combined, such as graph theory, information theory, and statistics.
In this project, we present methods for measuring Shannon’s entropy of graphs.
On algorithmic problems concerning graphs of higher degree of symmetrygraphhoc
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that
Aut GR = G depends very much on the description of the geometry with which one starts. For example, we
can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded
subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3
special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form.
The aim of this research is to develop an effective method for generation of graphs connected with classical
geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The
main approach is to avoid manual drawing and to calculate graph layout automatically according to its
formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it
becomes more complicated for graphs of geometrical nature. There are two main reasons for the
investigations of spectra: (1) very often spectra carry much more useful information about the graph than a
corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called
Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important
for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one
could find some new Ramanujan graphs.
ON ALGORITHMIC PROBLEMS CONCERNING GRAPHS OF HIGHER DEGREE OF SYMMETRYFransiskeran
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry.
We consider here k-valent plane and toroidal maps with faces of size a and b. The faces are said to be in a lego if the faces are organized in blocks that then tile the sphere. We expose some enumeration results and the technical enumeration methods.
Then we expose how we managed to draw the graphs from the combinatorial data.
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
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This presentation is about applications of graph theory applications....it is updated version it was given at international conference at applications of graph theory at KAULALAMPUR MALYSIA 2OO7
I am Dennis L. I am an Algorithm Design Exam Expert at programmingexamhelp.com. I hold a Ph.D. in Computer Science from, the City University of New York. I have been helping students with their exams for the past 7 years. You can hire me to take your exam in Algorithm Design.
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A NEW PARALLEL ALGORITHM FOR COMPUTING MINIMUM SPANNING TREEijscmc
Computing the minimum spanning tree of the graph is one of the fundamental computational problems. In
this paper, we present a new parallel algorithm for computing the minimum spanning tree of an undirected
weighted graph with n vertices and m edges. This algorithm uses the cluster techniques to reduce the
number of processors by fraction 1/f (n) and the parallel work by the fraction O ( 1 lo g ( f ( n )) ),where f (n) is an
arbitrary function. In the case f (n) =1, the algorithm runs in logarithmic-time and use super linear work on
EREWPRAM model. In general, the proposed algorithm is the simplest one.
A NEW PARALLEL ALGORITHM FOR COMPUTING MINIMUM SPANNING TREEijscmcj
Computing the minimum spanning tree of the graph is one of the fundamental computational problems. In this paper, we present a new parallel algorithm for computing the minimum spanning tree of an undirected weighted graph with vertices and edges. This algorithm uses the cluster techniques to reduce the number of processors by fraction and the parallel work by the fraction O ( 1 lo g ( f ( n )) ),where f (n) is an arbitrary function. In the case f (n) =1, the algorithm runs in logarithmic-time and use super linear work on EREWPRAM model. In general, the proposed algorithm is the simplest one.
I am Craig D. I am a Stochastic Processes Assignment Expert at statisticsassignmenthelp.com. I hold a Master's in Statistics, from The University of Queensland. I have been helping students with their homework for the past 9 years. I solve assignments related to Stochastic Processes.
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You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
I am Stacy W. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of McGill, Canada
I have been helping students with their homework for the past 7years. I solve assignments related to Statistical.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
I am Joe L. I am an Algorithms Design Assignment Expert at programminghomeworkhelp.com. I hold a Ph.D. in Programming from, the University of Chicago, USA. I have been helping students with their assignments for the past 10 years. I solve assignments related to Algorithms Design.
Visit programminghomeworkhelp.com or email support@programminghomeworkhelp.com. You can also call on +1 678 648 4277 for any assistance with the Algorithm Design Assignment.
Much of the early work on parameterized complexity considered the solution size as the parameter when parameterizing optimization problems, with a possible exception of treewidth. This talk will survey results and open problems on *alternate parameterizations*, where the parameter is typically some structure of the input or the distance of the output size from a guarantee.
Dynamic Parameterized Problems - Algorithms and Complexitycseiitgn
In this talk, we will discuss the parameterized complexity of various classical graph-theoretic problems in the dynamic framework where the input graph is being updated by a sequence of edge additions and deletions. We will describe fixed-parameter tractable algorithms and lower bounds on the running time of algorithms for these problems.
Space-efficient Approximation Scheme for Maximum Matching in Sparse Graphscseiitgn
We present a Logspace Approximation Scheme (LSAS), i.e. an approximation algorithm for maximum matching in planar graphs (not necessarily bipartite) that achieves an approximation ratio arbitrarily close to one, in Logspace This deviates from the well known Baker’s approach for approximation in planar graphs by avoiding the use of distance computation - which is not known to be in Logspace. Our algorithm actually works for any “recursively sparse” graph class which contains a linear size matching.The scheme is based on an LSAS in bounded degree graphs which are not known to be amenable to Baker’s method. We solve the bounded degree case by parallel augmentation of short augmenting paths. Finding a large number of such disjoint paths can, in turn, be reduced to finding a large independent set in a bounded degree graph. This is joint work with Samir Datta and Raghav Kulkarni.
A multiple choice problem consists of a set of color classes P = {C1 , C2 , . . . , Cn }. Each color class Ci consists of a pair of objects typically a pair of points. Objective of such a problem, is to select one object from each color class such that certain optimality criteria is satisfied. One example of such problem is rainbow minmax gap problem(RMGP). In RMGP, given P, the objective is to select exactly one point from each color class, such that the maximum distance between a pair of consecutive selected points is minimized. This problem was studied by Consuegra and Narasimhan. We show that the problem is NP-hard. For our proof we also describe an auxiliary result on satisfiability. A 3-SAT formula is an LSAT formula if each clause (viewed as a set of literals) intersects at most one other clause, and, moreover, if two clauses intersect, then they have exactly one literal in common. We show that the problem of deciding whether an LSAT formula is satisfiable or not is NP-complete. We also briefly describe some approximation results of some multiple choice problems.
The Chasm at Depth Four, and Tensor Rank : Old results, new insightscseiitgn
Agrawal and Vinay [FOCS 2008] showed how any polynomial size arithmetic circuit can be thought of as a depth four arithmetic circuit of subexponential size. The resulting circuit size in this simulation was more carefully analyzed by Koiran [TCS 2012] and subsequently by Tavenas [MFCS 2013]. We provide a simple proof of this chain of results. We then abstract the main ingredient to apply it to formulas and constant depth circuits, and show more structured depth reductions for them.In an apriori surprising result, Raz [STOC 2010] showed that for any $n$ and $d$, such that $\omega(1) \leq d \leq O(logn/loglogn)$, constructing explicit tensors $T: [n] \rightarrow F$ of high enough rank would imply superpolynomial lower bounds for arithmetic formulas over the field F. Using the additional structure we obtain from our proof of the depth reduction for arithmetic formulas, we give a new and arguably simpler proof of this connection. We also extend this result for homogeneous formulas to show that, in fact, the connection holds for any d such that $\omega(1) \leq d \leq n^{o(1)}$. Joint work with Mrinal Kumar, Ramprasad Saptharishi and V Vinay.
Isolation Lemma for Directed Reachability and NL vs. Lcseiitgn
We discuss versions of isolation lemma for problems in NP and NL. We then show that improving the success probability of the isolation lemma is equivalent to some complexity theoretic collapses such as NP is in P/poly and NL is in L/poly. Basic familiarity with complexity classes such as NP, P/poly, NL will be assumed. All the other notions used in the talk will be introduced during the talk.
Narrow sieves, representative sets and divide-and-color are three breakthrough techniques related to color coding, which led to the design of extremely fast parameterized algorithms. In this talk, I will discuss the power and limitations of these techniques. I will also briefly address some recent developments related to these techniques, including general schemes for mixing them.
Solving connectivity problems via basic Linear Algebracseiitgn
Directed reachability and undirected connectivity are well studied problems in Complexity Theory. Reachability/Connectivity between distinct pairs of vertices through disjoint paths are well known but hard variations. We talk about recent algorithms to solve variants and restrictions of these problems in the static and dynamic settings by reductions to the determinant.
We study communication cost of computing functions when inputs are distributed among k processors, each of which is located at one vertex of a network/graph called a terminal. Every other node of the network also has a processor, with no input. The communication is point-to-point and the cost is the total number of bits exchanged by the protocol, in the worst case, on all edges. Our results show the effect of topology of the network on the total communication cost. We prove tight bounds for simple functions like Element-Distinctness (ED), which depend on the 1-median of the graph. On the other hand, we show that for a large class of natural functions like Set-Disjointness the communication cost is essentially n times the cost of the optimal Steiner tree connecting the terminals. Further, we show for natural composed functions like ED∘XOR and XOR∘ED, the naive protocols suggested by their definition is optimal for general networks. Interestingly, the bounds for these functions depend on more involved topological parameters that are a combination of Steiner tree and 1-median costs. To obtain our results, we use some tools like metric embeddings and linear programming whose use in the context of communication complexity is novel as far as we know. (Based on joint works with Jaikumar Radhakrishnan and Atri Rudra)
Space-efficient Approximation Scheme for Maximum Matching in Sparse Graphscseiitgn
We present a Logspace Approximation Scheme (LSAS), i.e. an approximation algorithm for maximum matching in planar graphs (not necessarily bipartite) that achieves an approximation ratio arbitrarily close to one, in Logspace This deviates from the well known Baker’s approach for approximation in planar graphs by avoiding the use of distance computation - which is not known to be in Logspace. Our algorithm actually works for any “recursively sparse” graph class which contains a linear size matching.The scheme is based on an LSAS in bounded degree graphs which are not known to be amenable to Baker’s method. We solve the bounded degree case by parallel augmentation of short augmenting paths. Finding a large number of such disjoint paths can, in turn, be reduced to finding a large independent set in a bounded degree graph. This is joint work with Samir Datta and Raghav Kulkarni.
Efficiently decoding Reed-Muller codes from random errorscseiitgn
"Consider the following setting. Suppose we are given as input a ""corrupted"" truth-table of a polynomial f(x1,..,xm) of degree r = o(√m), with a random set of 1/2 - o(1) fraction of the evaluations flipped. Can the polynomial f be recovered? Turns out we can, and we can do it efficiently! The above question is a demonstration of the reliability of Reed-Muller codes under the random error model. For Reed-Muller codes over F_2, the message is thought of as an m-variate polynomial of degree r, and its encoding is just the evaluation over all of F_2^m.
In this talk, we shall study the resilience of RM codes in the *random error* model. We shall see that under random errors, RM codes can efficiently decode many more errors than its minimum distance. (For example, in the above toy example, minimum distance is about 1/2^{√m} but we can correct close to 1/2-fraction of random errors). This builds on a recent work of [Abbe-Shpilka-Wigderson-2015] who established strong connections between decoding erasures and decoding errors. The main result in this talk would be constructive versions of those connections that yield efficient decoding algorithms. This is joint work with Amir Shpilka and Ben Lee Volk."
Complexity Classes and the Graph Isomorphism Problemcseiitgn
The Graph Isomorphism problem is one of the few problems in NP, but not expected to be NP complete and not known to be in P.In this talk I will review some of the attempts that have been made in order to provide a better classification of the problem in terms of complexity classes reviewing upper and lower bounds and illustrating in this way the utility of several complexity classes.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
A Quest for Subexponential Time Parameterized Algorithms for Planar-k-Path: From Undirected to Directed
1. A Quest for Subexponential Time Parameterized
Algorithms for Planar-k-Path: From Undirected to
Directed Graphs
Saket Saurabh1
1Institute for Mathematical Sciences, Chennai, India,
and University of Bergen, Norway.
NMI Workshop on Complexity Theory, IITGN, Saturday 5th
November, 2016
(Slides by D. Marx, M. Pilipzuck and S. Saurabh)
1
2. Main message
NP-hard problems become easier on planar graphs
and geometric objects, and usually exactly by a
square root factor.
Planar graphs Geometric objects
2
3. Better exponential algorithms
Most NP-hard problems (e.g., 3-Coloring, Independent Set,
Hamiltonian Cycle, Steiner Tree, etc.) remain NP-hard on
planar graphs,1 so what do we mean by “easier”?
1
Notable exception: Max Cut is in P for planar graphs.
3
4. Better exponential algorithms
Most NP-hard problems (e.g., 3-Coloring, Independent Set,
Hamiltonian Cycle, Steiner Tree, etc.) remain NP-hard on
planar graphs,1 so what do we mean by “easier”?
The running time is still exponential, but significantly smaller:
2O(n)
) 2O(
p
n)
nO(k)
) nO(
p
k)
2O(k)
· nO(1)
) 2O(
p
k)
· nO(1)
1
Notable exception: Max Cut is in P for planar graphs.
3
6. Chapter 1: Subexponential algorithms using treewidth
Treewidth is a measure of “how treelike the graph is.”
We need only the following basic facts:
Treewidth
1 If a graph G has treewidth k, then many classical NP-hard
problems can be solved in time 2O(k) · nO(1) or
2O(k log k) · nO(1) on G.
2 A planar graph on n vertices has treewidth O(
p
n).
5
7. Treewidth — a measure of “tree-likeness”
Tree decomposition: Vertices are arranged in a tree structure
satisfying the following properties:
1 If u and v are neighbors, then there is a bag containing both
of them.
2 For every v, the bags containing v form a connected subtree.
Width of the decomposition: largest bag size 1.
treewidth: width of the best decomposition.
dcb
a
e f g h
g, hb, e, fa, b, c
d, f , gb, c, f
c, d, f
6
8. Treewidth — a measure of “tree-likeness”
Tree decomposition: Vertices are arranged in a tree structure
satisfying the following properties:
1 If u and v are neighbors, then there is a bag containing both
of them.
2 For every v, the bags containing v form a connected subtree.
Width of the decomposition: largest bag size 1.
treewidth: width of the best decomposition.
hgfe
a
b c d
g, hb, e, fa, b, c
d, f , gb, c, f
c, d, f
A subtree communicates with the outside world
only via the root of the subtree.
6
9. Finding tree decompositions
Various algorithms for finding optimal or approximate tree
decompositions if treewidth is w:
optimal decomposition in time 2O(w3) · n
[Bodlaender 1996].
4-approximate decomposition in time 2O(w) · n2
[Robertson and Seymour].
5-approximate decomposition in time 2O(w) · n
[Bodlaender et al. 2013].
O(
p
log w)-approximation in polynomial time
[Feige, Hajiaghayi, Lee 2008].
As we are mostly interested in algorithms with running time
2O(w) · nO(1), we may assume that we have a decomposition.
7
10. 3-Coloring and tree decompositions
Theorem
Given a tree decomposition of width w, 3-Coloring can be
solved in time 3w · wO(1) · n.
Bx : vertices appearing in node x.
Vx : vertices appearing in the subtree rooted at x.
For every node x and coloring c : Bx !
{1, 2, 3}, we compute the Boolean value
E[x, c], which is true if and only if c can
be extended to a proper 3-coloring of Vx .
Claim:
We can determine E[x, c] if all the values are
known for the children of x.
c, d, f
b, c, f d, f , g
a, b, c b, e, f g, h
bcf=T bcf=F
bcf=T bcf=F
. . . . . .
8
11. Subexponential algorithm for 3-Coloring
Theorem [textbook dynamic programming]
3-Coloring can be solved in time 2O(w) · nO(1) on graphs of
treewidth w.
+
Theorem [Robertson and Seymour]
A planar graph on n vertices has treewidth O(
p
n).
9
12. Subexponential algorithm for 3-Coloring
Theorem [textbook dynamic programming]
3-Coloring can be solved in time 2O(w) · nO(1) on graphs of
treewidth w.
+
Theorem [Robertson and Seymour]
A planar graph on n vertices has treewidth O(
p
n).
+
Corollary
3-Coloring can be solved in time 2O(
p
n) on planar graphs.
textbook algorithm + combinatorial bound
+
subexponential algorithm
9
13. Subexponential algorithm for 3-Coloring
Theorem
3-Coloring can be solved in time 2O(w) · nO(1) on graphs of
treewidth w.
+
Theorem [Robertson and Seymour]
A planar graph on n vertices has treewidth O(
p
n).
+
Corollary
3-Coloring can be solved in time 2O(
p
n) on planar graphs.
textbook algorithm + combinatorial bound
+
subexponential algorithm
10
14. Lower bounds
Corollary
3-Coloring can be solved in time 2O(
p
n) on planar graphs.
Two natural questions:
Can we achieve this running time on general graphs?
Can we achieve even better running time (e.g., 2O( 3p
n)) on
planar graphs?
11
15. Lower bounds
Corollary
3-Coloring can be solved in time 2O(
p
n) on planar graphs.
Two natural questions:
Can we achieve this running time on general graphs?
Can we achieve even better running time (e.g., 2O( 3p
n)) on
planar graphs?
P 6= NP is not a sufficiently strong hypothesis: it is compatible with
3SAT being solvable in time 2O(n1/1000) or even in time nO(log n).
We need a stronger hypothesis!
11
16. Exponential Time Hypothesis (ETH)
Hypothesis introduced by Impagliazzo, Paturi, and Zane:
Exponential Time Hypothesis (ETH) [consequence of]
There is no 2o(n)-time algorithm for n-variable 3SAT.
Note: current best algorithm is 1.30704n [Hertli 2011].
Note: an n-variable 3SAT formula can have m = ⌦(n3) clauses.
12
17. Exponential Time Hypothesis (ETH)
Hypothesis introduced by Impagliazzo, Paturi, and Zane:
Exponential Time Hypothesis (ETH) [consequence of]
There is no 2o(n)-time algorithm for n-variable 3SAT.
Note: current best algorithm is 1.30704n [Hertli 2011].
Note: an n-variable 3SAT formula can have m = ⌦(n3) clauses.
Are there algorithms that are subexponential in the size n + m of
the 3SAT formula?
12
18. Exponential Time Hypothesis (ETH)
Hypothesis introduced by Impagliazzo, Paturi, and Zane:
Exponential Time Hypothesis (ETH) [consequence of]
There is no 2o(n)-time algorithm for n-variable 3SAT.
Note: current best algorithm is 1.30704n [Hertli 2011].
Note: an n-variable 3SAT formula can have m = ⌦(n3) clauses.
Are there algorithms that are subexponential in the size n + m of
the 3SAT formula?
Sparsification Lemma [Impagliazzo, Paturi, Zane 2001]
There is a 2o(n)-time algorithm for n-variable 3SAT.
m
There is a 2o(n+m)-time algorithm for n-variable m-clause 3SAT.
12
19. Lower bounds based on ETH
ETH + Sparsification Lemma
There is no 2o(n+m)-time algorithm for n-variable m-clause 3SAT.
The textbook reduction from 3SAT to 3-Coloring:
3SAT formula
n variables
m clauses
)
Graph G
O(n + m) vertices
O(n + m) edges
Corollary
Assuming ETH, there is no 2o(n) algorithm for 3-Coloring on an
n-vertex graph G.
13
20. Lower bounds based on ETH
ETH + Sparsification Lemma
There is no 2o(n+m)-time algorithm for n-variable m-clause 3SAT.
The textbook reduction from 3SAT to 3-Coloring:
3SAT formula
n variables
m clauses
)
Graph G
O(m) vertices
O(m) edges
(we can assume n = O(m))
Corollary
Assuming ETH, there is no 2o(n) algorithm for 3-Coloring on an
n-vertex graph G.
13
21. Transfering bounds
There are polynomial-time reductions from, say, 3-Coloring to
many other problems such that the reduction increases the number
of vertices by at most a constant factor.
Consequence: Assuming ETH, there is no 2o(n) time algorithm on
n-vertex graphs for
Independent Set
Clique
Dominating Set
Vertex Cover
Hamiltonian Path
Feedback Vertex Set
. . .
14
22. Lower bounds based on ETH
What about 3-Coloring on planar graphs?
The textbook reduction from 3-Coloring to Planar
3-Coloring uses a “crossover gadget” with 4 external connectors:
In every 3-coloring of the gadget, opposite external connectors
have the same color.
Every coloring of the external connectors where the opposite
vertices have the same color can be extended to the whole
gadgets.
If two edges cross, replace them with a crossover gadget.
15
23. Lower bounds based on ETH
What about 3-Coloring on planar graphs?
The textbook reduction from 3-Coloring to Planar
3-Coloring uses a “crossover gadget” with 4 external connectors:
In every 3-coloring of the gadget, opposite external connectors
have the same color.
Every coloring of the external connectors where the opposite
vertices have the same color can be extended to the whole
gadgets.
If two edges cross, replace them with a crossover gadget.
15
24. Lower bounds based on ETH
What about 3-Coloring on planar graphs?
The textbook reduction from 3-Coloring to Planar
3-Coloring uses a “crossover gadget” with 4 external connectors:
In every 3-coloring of the gadget, opposite external connectors
have the same color.
Every coloring of the external connectors where the opposite
vertices have the same color can be extended to the whole
gadgets.
If two edges cross, replace them with a crossover gadget.
15
25. Lower bounds based on ETH
The reduction from 3-Coloring to Planar 3-Coloring
introduces O(1) new edge/vertices for each crossing.
A graph with m edges can be drawn with O(m2) crossings.
3SAT formula
n variables
m clauses
)
Graph G
O(m) vertices
O(m) edges
)
Planar graph G0
O(m2) vertices
O(m2) edges
Corollary
Assuming ETH, there is no 2o(
p
n) algorithm for 3-Coloring on
an n-vertex planar graph G.
(Essentially observed by [Cai and Juedes 2001])
16
26. Summary of Chapter 1
Streamlined way of obtaining tight upper and lower bounds for
planar problems.
Upper bound:
Standard bounded-treewidth algorithm + treewidth bound on
planar graphs give 2O(
p
n) time subexponential algorithms.
Lower bound:
Textbook NP-hardness proof with quadratic blow up + ETH
rule out 2o(
p
n) algorithms.
Works for Hamiltonian Cycle, Vertex Cover,
Independent Set, Feedback Vertex Set, Dominating
Set, Steiner Tree, . . .
17
27. Chapter 2: Grid minors and bidimensionality
More refined analysis of the running time: we express the running
time as a function of input size n and a parameter k.
Definition
A problem is fixed-parameter tractable (FPT) parameterized by
k if it can be solved in time f (k) · nO(1) for some computable
function f .
Examples of FPT problems:
Finding a vertex cover of size k.
Finding a feedback vertex set of size k.
Finding a path of length k.
Finding k vertex-disjoint triangles.
. . .
Note: these four problems have 2O(k) · nO(1) time algorithms, which
is best possible on general graphs.
18
28. W[1]-hardness
Negative evidence similar to NP-completeness. If a problem is
W[1]-hard, then the problem is not FPT unless FPT=W[1].
Some W[1]-hard problems:
Finding a clique/independent set of size k.
Finding a dominating set of size k.
Finding k pairwise disjoint sets.
. . .
For these problems, the exponent of n has to depend on k
(the running time is typically nO(k)).
19
29. Subexponential parameterized algorithms
What kind of upper/lower bounds we have for f (k)?
For most problems, we cannot expect a 2o(k) · nO(1) time
algorithm on general graphs.
(As this would imply a 2o(n) algorithm.)
For most problems, we cannot expect a 2o(
p
k) · nO(1) time
algorithm on planar graphs.
(As this would imply a 2o(
p
n) algorithm.)
20
30. Subexponential parameterized algorithms
What kind of upper/lower bounds we have for f (k)?
For most problems, we cannot expect a 2o(k) · nO(1) time
algorithm on general graphs.
(As this would imply a 2o(n) algorithm.)
For most problems, we cannot expect a 2o(
p
k) · nO(1) time
algorithm on planar graphs.
(As this would imply a 2o(
p
n) algorithm.)
However, 2O(
p
k) · nO(1) algorithms do exist for several
problems on planar graphs, even for some W[1]-hard problems.
Quick proofs via grid minors and bidimensionality.
[Demaine, Fomin, Hajiaghayi, Thilikos 2004]
Next: subexponential parameterized algorithm for k-Path.
20
31. Minors
Definition
Graph H is a minor of G (H G) if H can be obtained from G by
deleting edges, deleting vertices, and contracting edges.
deleting uv
vu w
u v
contracting uv
Note: length of the longest path in H is at most the length of the
longest path in G.
21
32. Planar Excluded Grid Theorem
Theorem [Robertson, Seymour, Thomas 1994]
Every planar graph with treewidth at least 5k has a k ⇥ k grid
minor.
Note: for general graphs, treewidth at least k100 (now even k19) or
so guarantees a k ⇥ k grid minor [Chekuri and Chuzhoy 2013] and
[Chuzhoy 2016]!
22
33. Bidimensionality for k-Path
Observation: If the treewidth of a planar graph G is at least 5
p
k
) It has a
p
k ⇥
p
k grid minor (Planar Excluded Grid Theorem)
) The grid has a path of length at least k.
) G has a path of length at least k.
23
34. Bidimensionality for k-Path
Observation: If the treewidth of a planar graph G is at least 5
p
k
) It has a
p
k ⇥
p
k grid minor (Planar Excluded Grid Theorem)
) The grid has a path of length at least k.
) G has a path of length at least k.
We use this observation to find a path of length at least k on
planar graphs:
If treewidth w of G is at least 5
p
k:
we answer “there is a path of length at
least k.”
If treewidth w of G is less than 5
p
k,
then we can solve the problem in time
2O(w) · nO(1) = 2O(
p
k) · nO(1).
23
35. Bidimensionality for k-Path
Observation: If the treewidth of a planar graph G is at least 5
p
k
) It has a
p
k ⇥
p
k grid minor (Planar Excluded Grid Theorem)
) The grid has a path of length at least k.
) G has a path of length at least k.
We use this observation to find a path of length at least k on
planar graphs:
Set w := 5
p
k.
Find an O(1)-approximate tree
decomposition.
If treewidth is at least w: we can
answer “there is a path of length at
least k.”
If we get a tree decomposition of
width O(w), then we can solve the
problem in time
2O(w)
· nO(1)
= 2O(
p
k)
· nO(1)
.
23
36. Bidimensionality
Definition
A graph invariant x(G) is minor-bidimensional if
x(G0) x(G) for every minor G0 of G, and
If Gk is the k ⇥ k grid, then x(Gk) ck2
(for some constant c > 0).
Examples: minimum vertex cover, length of the longest path,
feedback vertex set are minor-bidimensional.
24
37. Bidimensionality
Definition
A graph invariant x(G) is minor-bidimensional if
x(G0) x(G) for every minor G0 of G, and
If Gk is the k ⇥ k grid, then x(Gk) ck2
(for some constant c > 0).
Examples: minimum vertex cover, length of the longest path,
feedback vertex set are minor-bidimensional.
24
38. Bidimensionality
Definition
A graph invariant x(G) is minor-bidimensional if
x(G0) x(G) for every minor G0 of G, and
If Gk is the k ⇥ k grid, then x(Gk) ck2
(for some constant c > 0).
Examples: minimum vertex cover, length of the longest path,
feedback vertex set are minor-bidimensional.
24
39. Summary of Chapter 2
Tight bounds for minor-bidimensional planar problems.
Upper bound:
Standard bounded-treewidth algorithm + planar excluded grid
theorem give 2O(
p
k) · nO(1) time FPT algorithms.
Lower bound:
Textbook NP-hardness proof with quadratic blow up + ETH
rule out 2o(
p
n) time algorithms ) no 2o(
p
k) · nO(1) time
algorithm.
Variant of theory works for contraction-bidimensional problems,
e.g., Independent Set, Dominating Set.
25
40. Limits of bidimensionality?
Perhaps the most basic problem:
Subgraph Isomorphism
Given graphs H and G, decide if G has a subgraph isomorphic to
H.
26
41. Limits of bidimensionality?
Perhaps the most basic problem:
Subgraph Isomorphism
Given graphs H and G, decide if G has a subgraph isomorphic to
H.
Theorem [Eppstein 1999]
Subgraph Isomorphism for planar graphs can be solved in time
2O(k log k) · n for k := |V (H)|.
26
42. Limits of bidimensionality?
Perhaps the most basic problem:
Subgraph Isomorphism
Given graphs H and G, decide if G has a subgraph isomorphic to
H.
Question already asked several time over the last decades:
Does the square root phenomenon appear for Subgraph
Isomorphism on planar graphs?
26
43. Limits of bidimensionality?
Perhaps the most basic problem:
Subgraph Isomorphism
Given graphs H and G, decide if G has a subgraph isomorphic to
H.
Question already asked several time over the last decades:
Does the square root phenomenon appear for Subgraph
Isomorphism on planar graphs?
Assuming ETH, there is no 2o(k/ log k)nO(1) time algorithm for
general patterns.
[Bodlaender, Nederlof, van der Zanden; ICALP’16)]
26
44. Limits of bidimensionality?
Perhaps the most basic problem:
Subgraph Isomorphism
Given graphs H and G, decide if G has a subgraph isomorphic to
H.
Question already asked several time over the last decades:
Does the square root phenomenon appear for Subgraph
Isomorphism on planar graphs?
Assuming ETH, there is no 2o(k/ log k)nO(1) time algorithm for
general patterns.
[Bodlaender, Nederlof, van der Zanden; ICALP’16)]
There is a 2O(k/ log k)nO(1) time (randomized) algorithm for
connected patterns.
[At the end of the talk – hopefully!]
26
45. Limits of bidimensionality? [Counting problems]
Counting is harder than decision:
Counting version of easy problems:
not clear if they remain easy.
Counting version of hard problems:
not clear if we can keep the same running time.
27
46. Limits of bidimensionality? [Counting problems]
Counting is harder than decision:
Counting version of easy problems:
not clear if they remain easy.
Counting version of hard problems:
not clear if we can keep the same running time.
Working on counting problems is fun:
You can revisit fundamental, “well-understood” problems.
Requires a new set of lower bound techniques.
Requires new algorithmic techniques.
27
47. Bidimensionality and counting
Does not work for counting k-paths in a planar graph:
If treewidth w is O(
p
k): can be solved in time
2O(w)nO(1) = 2O(
p
k)nO(1) using dynamic programming.
If treewidth w is larger than c
p
k: answer is positive, but how
much exactly?
28
48. Bidimensionality and counting
Does not work for counting k-paths in a planar graph:
If treewidth w is O(
p
k): can be solved in time
2O(w)nO(1) = 2O(
p
k)nO(1) using dynamic programming.
If treewidth w is larger than c
p
k: answer is positive, but how
much exactly?
What about counting vertex covers of size k in a planar graph?
28
49. Bidimensionality and counting
Does not work for counting k-paths in a planar graph:
If treewidth w is O(
p
k): can be solved in time
2O(w)nO(1) = 2O(
p
k)nO(1) using dynamic programming.
If treewidth w is larger than c
p
k: answer is positive, but how
much exactly?
Works for counting vertex covers of size k in a planar graph:
If treewidth w is O(
p
k): can be solved in time
2O(w)nO(1) = 2O(
p
k)nO(1) using dynamic programming.
If treewidth w is larger than c
p
k: answer is 0.
28
50. Counting k-matching
Counting matchings can be significantly harder than finding a
matching!
Counting perfect matchings is #P-hard [Valiant 1979].
Counting matchings of size k is #W[1]-hard
[Curticapean 2013], [Curticapean and M. 2014].
Counting matchings of size k is FPT in planar graphs.
[Frick 2004]
Open question: Is there a 2O(
p
k) · nO(1) algorithm for counting k
matchings in planar graphs?
29
51. Counting k-matching
Counting matchings can be significantly harder than finding a
matching!
Counting perfect matchings in planar graphs is
polynomial-time solvable.
[Kasteleyn 1961], [Temperley and Fischer 1961].
Corollary: we can count matchings covering n k vertices in
time nO(k)
. . . but (assuming ETH) there is no f (k)no(k/ log k) time
algorithm [Curticapean and Xia 2015].
29
52. Chapter 3: Finding bounded-treewidth solutions
So far, we have exploited that the input has bounded treewidth
and used standard algorithms.
30
53. Chapter 3: Finding bounded-treewidth solutions
So far, we have exploited that the input has bounded treewidth
and used standard algorithms.
Change of viewpoint:
In many cases, we have to exploit instead that the solution has
bounded treewidth.
30
54. Chapter 3: Finding bounded-treewidth solutions
So far the way we have used treewidth is to find something (e.g.,
Hamiltonian cycle) in a large bounded-treewidth graph:
31
55. Chapter 3: Finding bounded-treewidth solutions
So far the way we have used treewidth is to find something (e.g.,
Hamiltonian cycle) in a large bounded-treewidth graph:
31
56. Chapter 3: Finding bounded-treewidth solutions
But we can also find small bounded-treewidth objects in an arbitrary
large graph.
G
H
Theorem [Alon, Yuster, Zwick 1994]
Given a graph H and weighted graph G, we can find a minimum
weight subgraph of G isomorphic to H in time 2O(|V (H)|) · nO(tw(H)).
31
57. Chapter 3: Finding bounded-treewidth solutions
But we can also find small bounded-treewidth objects in an arbitrary
large graph.
G
H
Theorem [Alon, Yuster, Zwick 1994]
Given a graph H and weighted graph G, we can find a minimum
weight subgraph of G isomorphic to H in time 2O(|V (H)|) · nO(tw(H)).
If the problem can be formulated as finding a graph of treewidth
O(
p
k), then we get an nO(
p
k) time algorithm.
31
58. Examples
Several examples:
Planar k-Terminal Cut
Improvement from nO(k) to 2O(k) · nO(
p
k).
Planar Strongly Connected Subgraph
Improvement from nO(k) to 2O(k log k) · nO(
p
k).
Subset TSP with k cities in a planar graph
Improvement from 2O(k) · nO(1) to 2O(
p
k log k) · nO(1).
Rectilinear Steiner Tree with n points in a plane.
Improvement from 2O(n) · nO(1) to 2O(
p
n log n) · nO(1).
32
59. TSP
TSP
Input: A set T of cities and a distance function d on T
Output: A tour on T with minimum total distance
Theorem [Held and Karp]
TSP with k cities can be solved in time 2k · nO(1).
Dynamic programming:
Let x(v, T0) be the minimum length of path from vstart to v
visiting all the cities T0 ✓ T.
33
60. Subset TSP on planar graphs
Assume that the cities correspond to a subset T of a planar graph
and distance is measured in this planar graph.
34
61. Subset TSP on planar graphs
Assume that the cities correspond to a subset T of a planar graph
and distance is measured in this planar graph.
Can be solved in time 2O(
p
n).
Can be solved in time 2k · nO(1).
Question: Can we solve it in time 2O(
p
k) · nO(1)?
34
62. Subset TSP on planar graphs
Assume that the cities correspond to a subset T of a planar graph
and distance is measured in this planar graph.
Theorem [Klein and Marx]
Subset TSP for k cities in a (unit-weight) planar graph can be
solved in time 2O(
p
k log k) · nO(1).
34
63. TSP and treewidth
We wanted to formulate the problem as finding a low
treewidth subgraph.
A cycle has treewidth 2, is this of any help?
Problem:
We have to remember the subset of cities visited by the partial tour
(2k possibilities).
35
64. c-change TSP
c-change operation: removing c steps of the tour and
connecting the resulting c paths in some other way.
A solution is c-OPT if no c-change can improve it.
We can find a c-OPT solution in kO(c) · D time, where D is
the maximum distance (if distances are integers).
36
65. c-change TSP
c-change operation: removing c steps of the tour and
connecting the resulting c paths in some other way.
A solution is c-OPT if no c-change can improve it.
We can find a c-OPT solution in kO(c) · D time, where D is
the maximum distance (if distances are integers).
36
66. c-change TSP
c-change operation: removing c steps of the tour and
connecting the resulting c paths in some other way.
A solution is c-OPT if no c-change can improve it.
We can find a c-OPT solution in kO(c) · D time, where D is
the maximum distance (if distances are integers).
36
67. The treewidth bound
Consider the union of an optimum solution and a 4-OPT solution
as a graph on k vertices:
Lemma
The union of an optimum solution and a 4-OPT solution has
treewidth O(
p
k) [some techincal details omitted].
37
68. The treewidth bound
Lemma
The union of an optimum solution and a 4-OPT solution has
treewidth O(
p
k) [some techincal details omitted].
The union has separators of size O(
p
k).
In each component, the set of cities visited by the optimum
solution is nice: it is the same as what O(
p
k) segments of the
4-OPT tour visited (kO(
p
k) possibilities).
38
69. The treewidth bound
Lemma
The union of an optimum solution and a 4-OPT solution has
treewidth O(
p
k) [some techincal details omitted].
The union has separators of size O(
p
k).
In each component, the set of cities visited by the optimum
solution is nice: it is the same as what O(
p
k) segments of the
4-OPT tour visited (kO(
p
k) possibilities).
Use board for few intuition!
38
70. Steiner Tree
Steiner Tree
Input: A graph G, a set T of n terminal vertices, a weight function
w : E(G) ! Q 0
Parameter: |T|
Question: A minimum weight Steiner tree for T in G.
39
71. Steiner Tree
Dreyfus-Wagner gave an algorithm with running time
3n · log W · |V (G)|O(1). W is the maximum weight on an edge.
40
72. Steiner Tree
Dreyfus-Wagner gave an algorithm with running time
3n · log W · |V (G)|O(1). W is the maximum weight on an edge.
In 2013, Nederlof et. al. gave an alternative algorithm with
running time 2n · |V (G)|O(1) · W .
40
73. Steiner Tree
Dreyfus-Wagner gave an algorithm with running time
3n · log W · |V (G)|O(1). W is the maximum weight on an edge.
In 2013, Nederlof et. al. gave an alternative algorithm with
running time 2n · |V (G)|O(1) · W .
Approximation algorithm: Byrka et. al. gave an algorithm with
approximation factor ln(4) + ✏.
40
74. Rectilinear Steiner Tree
Rectilinear Steiner Tree
Input: A set T of n terminal points, recdistG or the taxi-
cab/rectilinear metric.
Parameter: |T| = n
Question: A minimum length network connecting any two points
in T.
41
78. Outline of the algorithm
Step 1: Interpret the problem as a Steiner Tree problem on a
grid graph.
45
79. Outline of the algorithm
Step 1: Interpret the problem as a Steiner Tree problem on a
grid graph.
Step 2: Find a 2-approximate solution bS for Rectilinear Steiner
Tree.
45
80. Outline of the algorithm
Step 1: Interpret the problem as a Steiner Tree problem on a
grid graph.
Step 2: Find a 2-approximate solution bS for Rectilinear Steiner
Tree.
Step 3: Show that there exists an optimal solution Sopt such
that the subgraph S = bS [ Sopt has a “useful” structure.
45
81. Outline of the algorithm
Step 1: Interpret the problem as a Steiner Tree problem on a
grid graph.
Step 2: Find a 2-approximate solution bS for Rectilinear Steiner
Tree.
Step 3: Show that there exists an optimal solution Sopt such
that the subgraph S = bS [ Sopt has a “useful” structure.
Step 4: Use a “ghost” structure to do dynamic programming
to solve the problem.
45
85. Step 1: Graphic Interpretation
Rectilinear Steiner Tree
Input: A set T of n terminal points, the Hanan grid G of T and
recdistG .
Parameter: |T| = n
Question: A minimum Steiner tree for T in G.
A Hanan grid with n terminals has at most n2 vertices in total.
49
89. Shortest path RST
Can ensure that at each iterative step, at most one bend
vertex is created. Total of n bend vertices.
53
90. Shortest path RST
Can ensure that at each iterative step, at most one bend
vertex is created. Total of n bend vertices.
The tree is constructed in polynomial time.
53
91. Outline of the algorithm
Step 1: Interpret the problem as a Steiner Tree problem on a
grid graph.
Step 2: Find a 2-approximate solution bS for Rectilinear Steiner
Tree.
Step 3: Show that there exists an optimal solution Sopt such
that the subgraph S = bS [ Sopt has a “useful” structure.
Step 4: Use a “ghost” structure to do dynamic programming
to solve the problem.
54
92. Outline of the algorithm
Step 1: Interpret the problem as a Steiner Tree problem on a
Hanan grid.
Step 2: Find a 2-approximate solution bS for Rectilinear Steiner
Tree.
Step 3: Show that there exists an optimal solution Sopt such
that the subgraph S = bS [ Sopt has a “useful” structure.
Step 4: Use a “ghost” structure to do dynamic programming
to solve the problem.
55
93. Outline of the algorithm
Step 1: Interpret the problem as a Steiner Tree problem on a
Hanan grid.
Step 2: Find a Shortest Path RST bS.
Step 3: Show that there exists an optimal solution Sopt such
that the subgraph S = bS [ Sopt has a “useful” structure.
Step 4: Use a “ghost” structure to do dynamic programming
to solve the problem.
56
94. Outline of the algorithm
Step 1: Interpret the problem as a Steiner Tree problem on a
Hanan grid.
Step 2: Find a Shortest Path RST bS.
Step 3: Show that there exists an optimal solution Sopt such
that the subgraph S = bS [ Sopt has bounded treewidth.
Step 4: Use a “ghost” structure to do dynamic programming
to solve the problem.
57
95. Outline of the algorithm
Step 1: Interpret the problem as a Steiner Tree problem on a
Hanan grid.
Step 2: Find a Shortest Path RST bS.
Step 3: Show that there exists an optimal solution Sopt such
that the subgraph S = bS [ Sopt has bounded treewidth.
Step 4: Use a “ghost” tree decomposition to do dynamic
programming to solve the problem.
58
96. Step 3: Bounded treewidth subgrid
Let Sopt be the optimal solution that has minimum number of
bends.
Sopt has 3n bend vertices.
59
97. Step 3: Bounded treewidth subgrid
Let Sopt be the optimal solution that has minimum number of
bends.
Sopt has 3n bend vertices.
bS has n bend vertices.
59
98. Step 3: Bounded treewidth subgrid
Let Sopt be the optimal solution that has minimum number of
bends.
Sopt has 3n bend vertices.
bS has n bend vertices.
There are n terminal vertices.
59
99. Step 3: Bounded treewidth subgrid
Attempt 1:
S = bS [ Sopt is a planar graph.
60
100. Step 3: Bounded treewidth subgrid
Attempt 1:
S = bS [ Sopt is a planar graph.
Has treewidth at most
p
|V (S)|.
60
101. Step 3: Bounded treewidth subgrid
Attempt 1:
S = bS [ Sopt is a planar graph.
Has treewidth at most
p
|V (S)|.
In this way, bound on treewidth as bad as O(n).
60
102. Step 3: Bounded treewidth subgrid
Theorem [Robertson, Seymour, Thomas 1994]
Every planar graph with treewidth at least 5k has a k ⇥ k grid
minor.
61
103. Step 3: Bounded treewidth subgrid
Theorem [Robertson, Seymour, Thomas 1994]
Every planar graph with treewidth at least 5k has a k ⇥ k grid
minor.
Suppose S has treewidth at most ↵
p
n. Then there is a
9
p
n ⇥ 9
p
n grid as a minor.
61
114. Steiner Tree Treewidth DP
The important step in the algorithm for Steiner Tree is to compute
the following information:
70
115. Steiner Tree Treewidth DP
The important step in the algorithm for Steiner Tree is to compute
the following information: for each bag Xt, X ✓ Xt and a partition
P = (P1, . . . , Pq) of X, the value c[t, X, P] is the minimum weight
of a subgraph F of St with the following properties:
70
116. Steiner Tree Treewidth DP
The important step in the algorithm for Steiner Tree is to compute
the following information: for each bag Xt, X ✓ Xt and a partition
P = (P1, . . . , Pq) of X, the value c[t, X, P] is the minimum weight
of a subgraph F of St with the following properties:
1 F has exactly q connected components C1, . . . , Cq such that
; 6= Pi = Xt V (Ci ) for all i 2 [q]. That is, P corresponds to
connected components of F.
70
117. Steiner Tree Treewidth DP
The important step in the algorithm for Steiner Tree is to compute
the following information: for each bag Xt, X ✓ Xt and a partition
P = (P1, . . . , Pq) of X, the value c[t, X, P] is the minimum weight
of a subgraph F of St with the following properties:
1 F has exactly q connected components C1, . . . , Cq such that
; 6= Pi = Xt V (Ci ) for all i 2 [q]. That is, P corresponds to
connected components of F.
2 Xt V (F) = X. That is, the vertices of Xt X are untouched
by F.
70
118. Steiner Tree Treewidth DP
The important step in the algorithm for Steiner Tree is to compute
the following information: for each bag Xt, X ✓ Xt and a partition
P = (P1, . . . , Pq) of X, the value c[t, X, P] is the minimum weight
of a subgraph F of St with the following properties:
1 F has exactly q connected components C1, . . . , Cq such that
; 6= Pi = Xt V (Ci ) for all i 2 [q]. That is, P corresponds to
connected components of F.
2 Xt V (F) = X. That is, the vertices of Xt X are untouched
by F.
3 T Vt ✓ V (F). That is, all the terminal vertices in St belong
to F.
70
119. Step 4: Treewidth Dynamic programming
In our case, we define types. Bags of a tree decomposition '
types.A type is a tuple (Y , Y 0 ✓ Y , P, T0) such that the following
holds.
71
120. Step 4: Treewidth Dynamic programming
In our case, we define types. Bags of a tree decomposition '
types.A type is a tuple (Y , Y 0 ✓ Y , P, T0) such that the following
holds.
(i) Y is a subset of V (G) of size at most q = 41
p
n + 2.
(ii) P is a partition of Y 0.
71
121. Step 4: Treewidth Dynamic programming
In our case, we define types. Bags of a tree decomposition '
types.A type is a tuple (Y , Y 0 ✓ Y , P, T0) such that the following
holds.
(i) Y is a subset of V (G) of size at most q = 41
p
n + 2.
(ii) P is a partition of Y 0.
(iii) There exists a set of components C1, . . . , Cq in bS Y such
that T0 = T Y 0 [
Sq
i=1 V (Ci ) .
71
122. Step 4: Treewidth Dynamic programming
There are at most 2
p
n log n types. This helps to design a dynamic
programming for the final step of the algorithm.
72
123. Summary of Chapter 3
Parameterized problems where bidimensionality does not work.
Upper bounds:
Algorithms based on finding a bounded-treewidth subgraph.
Treewidth bound is problem-specific:
k-Terminal Cut: dual solution has O(k) branch vertices.
Planar Strongly Connected Subgraph: solution
consists of O(k) paths/strips.
Subset TSP on planar graphs: the union of an optimum
solution and a 4-OPT solution has treewidth O(
p
k).
Rectilinear Steiner Tree on planar graphs: the union of
an optimum solution and an approximation solution has
treewidth O(
p
k).
Lower bounds:
To rule out f (k) · no(
p
k) time algorithms, we have to prove
W[1]-hardness by reduction from Grid Tiling.
73
125. Grid minors and treewidth
Grid Minor Theorem Robertson,
Seymour; GM V
There is a function f such that tw(G) f (k) implies the existence
of a k ⇥ k grid minor in G.
75
126. Grid minors and treewidth
Grid Minor Theorem Robertson,
Seymour; GM V
There is a function f such that tw(G) f (k) implies the existence
of a k ⇥ k grid minor in G.
Upper bound: f (k) 2 ˜O(k19) (Chuzhoy ’15)
75
127. Grid minors and treewidth
Grid Minor Theorem Robertson,
Seymour; GM V
There is a function f such that tw(G) f (k) implies the existence
of a k ⇥ k grid minor in G.
Upper bound: f (k) 2 ˜O(k19) (Chuzhoy ’15)
Lower bound: f (k) 2 ⌦(k2 log k)
75
128. Grid minors and treewidth
Grid Minor Theorem Robertson,
Seymour; GM V
There is a function f such that tw(G) f (k) implies the existence
of a k ⇥ k grid minor in G.
Upper bound: f (k) 2 ˜O(k19) (Chuzhoy ’15)
Lower bound: f (k) 2 ⌦(k2 log k)
Minor-free: f (k) 2 ⇥(k) for graphs excluding a fixed minor.
75
129. Grid minors and treewidth
Grid Minor Theorem Robertson,
Seymour; GM V
There is a function f such that tw(G) f (k) implies the existence
of a k ⇥ k grid minor in G.
Upper bound: f (k) 2 ˜O(k19) (Chuzhoy ’15)
Lower bound: f (k) 2 ⌦(k2 log k)
Minor-free: f (k) 2 ⇥(k) for graphs excluding a fixed minor.
Planar: tw(G) > 4.5k + 1 implies a k ⇥ k grid minor.
75
130. Grid minors and treewidth
Grid Minor Theorem Robertson,
Seymour; GM V
There is a function f such that tw(G) f (k) implies the existence
of a k ⇥ k grid minor in G.
Upper bound: f (k) 2 ˜O(k19) (Chuzhoy ’15)
Lower bound: f (k) 2 ⌦(k2 log k)
Minor-free: f (k) 2 ⇥(k) for graphs excluding a fixed minor.
Planar: tw(G) > 4.5k + 1 implies a k ⇥ k grid minor.
H-minor-free: tw(G) > cH · k implies a k ⇥ k grid minor.
75
133. Bidimensionality
k-Path: Is there a simple path on k vertices in a graph?
Goal: 2O(
p
k) · n algorithm for planar graphs.
Fact 1: k-Path can be solved in time 2O(w) · n on a tree
decomposition of width w.
76
134. Bidimensionality
k-Path: Is there a simple path on k vertices in a graph?
Goal: 2O(
p
k) · n algorithm for planar graphs.
Fact 1: k-Path can be solved in time 2O(w) · n on a tree
decomposition of width w.
Fact 2: If there is
p
k ⇥
p
k grid minor, then there is a k-path.
76
137. Algorithm
Approximate treewidth.
If tw 100
p
k then there is a
p
k ⇥
p
k grid minor, so also a
k-path.
Otherwise we have a tree decomposition of width O(
p
k).
Apply dynamic programming.
77
140. Bidimensionality: drawbacks
The argument is elegant, but very delicate.
Assumption “large grid minor ) certain answer” is very strong.
Directed k-Path: simple path on k vertices in a
directed graph.
78
141. Bidimensionality: drawbacks
The argument is elegant, but very delicate.
Assumption “large grid minor ) certain answer” is very strong.
Directed k-Path: simple path on k vertices in a
directed graph.
A large grid minor in the underlying undirected graph tells
little about directed k-paths.
78
142. Bidimensionality: drawbacks
The argument is elegant, but very delicate.
Assumption “large grid minor ) certain answer” is very strong.
Directed k-Path: simple path on k vertices in a
directed graph.
A large grid minor in the underlying undirected graph tells
little about directed k-paths.
Similar problem for searching for k-path of maximum weight,
or a cycle of length exactly k, ...
78
143. Bidimensionality: drawbacks
The argument is elegant, but very delicate.
Assumption “large grid minor ) certain answer” is very strong.
Directed k-Path: simple path on k vertices in a
directed graph.
A large grid minor in the underlying undirected graph tells
little about directed k-paths.
Similar problem for searching for k-path of maximum weight,
or a cycle of length exactly k, ...
Can such problems be solved in time 2
˜O(
p
k) · nc on planar
graphs?
78
144. Bidimensionality: drawbacks
The argument is elegant, but very delicate.
Assumption “large grid minor ) certain answer” is very strong.
Directed k-Path: simple path on k vertices in a
directed graph.
A large grid minor in the underlying undirected graph tells
little about directed k-paths.
Similar problem for searching for k-path of maximum weight,
or a cycle of length exactly k, ...
Can such problems be solved in time 2
˜O(
p
k) · nc on planar
graphs?
Until Recently, no better algorithms were known than
2O(k)
· nc
inherited from the general setting.
78
145. Bidimensionality: drawbacks
The argument is elegant, but very delicate.
Assumption “large grid minor ) certain answer” is very strong.
Directed k-Path: simple path on k vertices in a
directed graph.
A large grid minor in the underlying undirected graph tells
little about directed k-paths.
Similar problem for searching for k-path of maximum weight,
or a cycle of length exactly k, ...
Can such problems be solved in time 2
˜O(
p
k) · nc on planar
graphs?
Until Recently, no better algorithms were known than
2O(k)
· nc
inherited from the general setting.
General: Let a k-pattern be a vertex subset P such that
|P| k and G[P] is connected.
78
146. Bidimensionality: drawbacks
The argument is elegant, but very delicate.
Assumption “large grid minor ) certain answer” is very strong.
Directed k-Path: simple path on k vertices in a
directed graph.
A large grid minor in the underlying undirected graph tells
little about directed k-paths.
Similar problem for searching for k-path of maximum weight,
or a cycle of length exactly k, ...
Can such problems be solved in time 2
˜O(
p
k) · nc on planar
graphs?
Until Recently, no better algorithms were known than
2O(k)
· nc
inherited from the general setting.
General: Let a k-pattern be a vertex subset P such that
|P| k and G[P] is connected.
We are looking with k-patterns with some prescribed property.
78
147. Bidimensionality: drawbacks
The argument is elegant, but very delicate.
Assumption “large grid minor ) certain answer” is very strong.
Directed k-Path: simple path on k vertices in a
directed graph.
A large grid minor in the underlying undirected graph tells
little about directed k-paths.
Similar problem for searching for k-path of maximum weight,
or a cycle of length exactly k, ...
Can such problems be solved in time 2
˜O(
p
k) · nc on planar
graphs?
Until Recently, no better algorithms were known than
2O(k)
· nc
inherited from the general setting.
General: Let a k-pattern be a vertex subset P such that
|P| k and G[P] is connected.
We are looking with k-patterns with some prescribed property.
Idea: Find a small family of subgraphs of small treewidth that
cover every pattern.
78
148. Pattern coverage
Main result
Suppose G is a planar graph on n vertices, and suppose k is a
positive integer. Then there exists a family F of subsets of V (G)
such that:
(a) |F| 2
˜O(
p
k) · nc for some constant c;
(b) For each A 2 F, we have tw(G[A]) ˜O(
p
k);
(c) For every k-pattern P, there is some A 2 F such that P ✓ A.
Moreover, F can be constructed in randomized time 2
˜O(
p
k) · nc.
79
149. Pattern coverage
Main result
Suppose G is a planar graph on n vertices, and suppose k is a
positive integer. Then there exists a family F of subsets of V (G)
such that:
(a) |F| 2
˜O(
p
k) · nc for some constant c;
(b) For each A 2 F, we have tw(G[A]) ˜O(
p
k);
(c) For every k-pattern P, there is some A 2 F such that P ✓ A.
Moreover, F can be constructed in randomized time 2
˜O(
p
k) · nc.
Directed k-Path algorithm:
79
150. Pattern coverage
Main result
Suppose G is a planar graph on n vertices, and suppose k is a
positive integer. Then there exists a family F of subsets of V (G)
such that:
(a) |F| 2
˜O(
p
k) · nc for some constant c;
(b) For each A 2 F, we have tw(G[A]) ˜O(
p
k);
(c) For every k-pattern P, there is some A 2 F such that P ✓ A.
Moreover, F can be constructed in randomized time 2
˜O(
p
k) · nc.
Directed k-Path algorithm:
Construct F.
79
151. Pattern coverage
Main result
Suppose G is a planar graph on n vertices, and suppose k is a
positive integer. Then there exists a family F of subsets of V (G)
such that:
(a) |F| 2
˜O(
p
k) · nc for some constant c;
(b) For each A 2 F, we have tw(G[A]) ˜O(
p
k);
(c) For every k-pattern P, there is some A 2 F such that P ✓ A.
Moreover, F can be constructed in randomized time 2
˜O(
p
k) · nc.
Directed k-Path algorithm:
Construct F.
For each A 2 F, try to find a k-path in G[A] by DP.
79
152. Pattern coverage
Main result
Suppose G is a planar graph on n vertices, and suppose k is a
positive integer. Then there exists a family F of subsets of V (G)
such that:
(a) |F| 2
˜O(
p
k) · nc for some constant c;
(b) For each A 2 F, we have tw(G[A]) ˜O(
p
k);
(c) For every k-pattern P, there is some A 2 F such that P ✓ A.
Moreover, F can be constructed in randomized time 2
˜O(
p
k) · nc.
Directed k-Path algorithm:
Construct F.
For each A 2 F, try to find a k-path in G[A] by DP.
Running time: (2
˜O(
p
k)
· nc
) · (2
˜O(
p
k)
· n) = 2
˜O(
p
k)
· nc+1
79
154. Applications
Applies to a range of pattern-searching problems:
(directed) k-path of largest weight;
80
155. Applications
Applies to a range of pattern-searching problems:
(directed) k-path of largest weight;
cycle of length exactly k;
80
156. Applications
Applies to a range of pattern-searching problems:
(directed) k-path of largest weight;
cycle of length exactly k;
k-local search for Planar Vertex Cover.
80
157. Applications
Applies to a range of pattern-searching problems:
(directed) k-path of largest weight;
cycle of length exactly k;
k-local search for Planar Vertex Cover.
Needed: 2
˜O(w)
· nc
algorithm on graphs of treewidth w.
80
158. Applications
Applies to a range of pattern-searching problems:
(directed) k-path of largest weight;
cycle of length exactly k;
k-local search for Planar Vertex Cover.
Needed: 2
˜O(w)
· nc
algorithm on graphs of treewidth w.
Subgraph Isomorphism:
Given G and H with |V (H)| k, is H a subgraph of G?
80
159. Applications
Applies to a range of pattern-searching problems:
(directed) k-path of largest weight;
cycle of length exactly k;
k-local search for Planar Vertex Cover.
Needed: 2
˜O(w)
· nc
algorithm on graphs of treewidth w.
Subgraph Isomorphism:
Given G and H with |V (H)| k, is H a subgraph of G?
When H is connected and has maximum degree bounded by a
constant, we obtain an algorithm with running time 2
˜O(
p
k)
·nc
.
80
160. Applications
Applies to a range of pattern-searching problems:
(directed) k-path of largest weight;
cycle of length exactly k;
k-local search for Planar Vertex Cover.
Needed: 2
˜O(w)
· nc
algorithm on graphs of treewidth w.
Subgraph Isomorphism:
Given G and H with |V (H)| k, is H a subgraph of G?
When H is connected and has maximum degree bounded by a
constant, we obtain an algorithm with running time 2
˜O(
p
k)
·nc
.
Without the maxdeg bound, we get running time
2O(k/ log k)
· nc
using (Bodlaender, Nederlof, van der Zanden;
ICALP’16).
80
161. Applications
Applies to a range of pattern-searching problems:
(directed) k-path of largest weight;
cycle of length exactly k;
k-local search for Planar Vertex Cover.
Needed: 2
˜O(w)
· nc
algorithm on graphs of treewidth w.
Subgraph Isomorphism:
Given G and H with |V (H)| k, is H a subgraph of G?
When H is connected and has maximum degree bounded by a
constant, we obtain an algorithm with running time 2
˜O(
p
k)
·nc
.
Without the maxdeg bound, we get running time
2O(k/ log k)
· nc
using (Bodlaender, Nederlof, van der Zanden;
ICALP’16).
This running time is tight under ETH! (Bodlaender et al.)
80
163. Example: grid
Take a k ⇥ k grid.
Baker: Guess an index modulo
p
k s.t. there are
p
k
vertices of the pattern in the corresponding rows.
81
164. Example: grid
Take a k ⇥ k grid.
Baker: Guess an index modulo
p
k s.t. there are
p
k
vertices of the pattern in the corresponding rows.
Guess columns in which these vertices lie, kp
k options.
81
165. Example: grid
Take a k ⇥ k grid.
Baker: Guess an index modulo
p
k s.t. there are
p
k
vertices of the pattern in the corresponding rows.
Guess columns in which these vertices lie, kp
k options.
Remove the rows apart from intersections with columns.
What remains has treewidth O(
p
k).
81
166. Example: grid
Take a k ⇥ k grid.
Baker: Guess an index modulo
p
k s.t. there are
p
k
vertices of the pattern in the corresponding rows.
Guess columns in which these vertices lie, kp
k options.
Remove the rows apart from intersections with columns.
What remains has treewidth O(
p
k).
In total,
p
k · kp
k = 2O(
p
k log k) possible guesses.
81
167. Glimpse into the proof
Randomized statement:
There is a polytime algorithm that samples A with treewidth
˜O(
p
k) s.t. every k-pattern is covered with probability
(2
˜O(
p
k) · nc) 1.
82
168. Glimpse into the proof
Randomized statement:
There is a polytime algorithm that samples A with treewidth
˜O(
p
k) s.t. every k-pattern is covered with probability
(2
˜O(
p
k) · nc) 1.
Idea: Emulate a recursive approximation algorithm for
treewidth, trying to construct a decomposition of width
p
k.
82
169. Glimpse into the proof
Randomized statement:
There is a polytime algorithm that samples A with treewidth
˜O(
p
k) s.t. every k-pattern is covered with probability
(2
˜O(
p
k) · nc) 1.
Idea: Emulate a recursive approximation algorithm for
treewidth, trying to construct a decomposition of width
p
k.
When the algorithm gets stuck, apply Baker-like sparsification.
82
170. Glimpse into the proof
Randomized statement:
There is a polytime algorithm that samples A with treewidth
˜O(
p
k) s.t. every k-pattern is covered with probability
(2
˜O(
p
k) · nc) 1.
Idea: Emulate a recursive approximation algorithm for
treewidth, trying to construct a decomposition of width
p
k.
When the algorithm gets stuck, apply Baker-like sparsification.
State of the algorithm:
Graph G with ˜O(
p
k) terminals on the outer face.
82
171. Glimpse into the proof
Randomized statement:
There is a polytime algorithm that samples A with treewidth
˜O(
p
k) s.t. every k-pattern is covered with probability
(2
˜O(
p
k) · nc) 1.
Idea: Emulate a recursive approximation algorithm for
treewidth, trying to construct a decomposition of width
p
k.
When the algorithm gets stuck, apply Baker-like sparsification.
State of the algorithm:
Graph G with ˜O(
p
k) terminals on the outer face.
Pieces of the (unkown) pattern “hang” from terminals.
82
173. Proof strategy
Observation: Everything would be trivial if the pattern would
be within an ˜O(
p
k) margin from the outer face.
Otherwise, one of three possibilities holds:
˜O(
p
k)
83
174. Proof strategy
Observation: Everything would be trivial if the pattern would
be within an ˜O(
p
k) margin from the outer face.
Otherwise, one of three possibilities holds:
1 Balanced separator that avoids the pattern outside the margin.
83
175. Proof strategy
Observation: Everything would be trivial if the pattern would
be within an ˜O(
p
k) margin from the outer face.
Otherwise, one of three possibilities holds:
1 Balanced separator that avoids the pattern outside the margin.
Otherwise we “locate” a vertex z of the pattern outside the
margin.
z
83
176. Proof strategy
Observation: Everything would be trivial if the pattern would
be within an ˜O(
p
k) margin from the outer face.
Otherwise, one of three possibilities holds:
1 Balanced separator that avoids the pattern outside the margin.
Otherwise we “locate” a vertex z of the pattern outside the
margin.
2 Many concentric short separators separating z from the outer
face.
z
83
177. Proof strategy
Observation: Everything would be trivial if the pattern would
be within an ˜O(
p
k) margin from the outer face.
Otherwise, one of three possibilities holds:
1 Balanced separator that avoids the pattern outside the margin.
Otherwise we “locate” a vertex z of the pattern outside the
margin.
2 Many concentric short separators separating z from the outer
face.
3 Many “almost” disjoint paths connecting z with the outer face.
83
180. Achieving progress
Balanced separator: Remove outside margin and split.
Progress: The graph size and the pattern size are halved.
Concentric short separators:
z
84
181. Achieving progress
Balanced separator: Remove outside margin and split.
Progress: The graph size and the pattern size are halved.
Concentric short separators:
Progress: Guess a separator that splits the pattern in a
balanced way, and has a small intersection with it.
z
84
182. Achieving progress
Balanced separator: Remove outside margin and split.
Progress: The graph size and the pattern size are halved.
Concentric short separators:
Progress: Guess a separator that splits the pattern in a
balanced way, and has a small intersection with it.
Nearly disjoint paths:
84
183. Achieving progress
Balanced separator: Remove outside margin and split.
Progress: The graph size and the pattern size are halved.
Concentric short separators:
Progress: Guess a separator that splits the pattern in a
balanced way, and has a small intersection with it.
Nearly disjoint paths:
Progress: Guess a path where private are not in the pattern.
84
184. Achieving progress
Balanced separator: Remove outside margin and split.
Progress: The graph size and the pattern size are halved.
Concentric short separators:
Progress: Guess a separator that splits the pattern in a
balanced way, and has a small intersection with it.
Nearly disjoint paths:
Progress: Guess a path where private are not in the pattern.
Contract along it, thus dragging more of the pattern into the
margin.
84
185. Achieving progress
Balanced separator: Remove outside margin and split.
Progress: The graph size and the pattern size are halved.
Concentric short separators:
Progress: Guess a separator that splits the pattern in a
balanced way, and has a small intersection with it.
Nearly disjoint paths:
Progress: Guess a path where private are not in the pattern.
Contract along it, thus dragging more of the pattern into the
margin.
Keep track of 3 potentials measuring the progress to estimate
the success probability.
84
187. Generalizations
We rely on two properties of the graph class C:
Minor-closed: If H G and G 2 C, then H 2 C.
85
188. Generalizations
We rely on two properties of the graph class C:
Minor-closed: If H G and G 2 C, then H 2 C.
Locally bounded treewidth: tw(G) f (rad(G)) for some f .
85
189. Generalizations
We rely on two properties of the graph class C:
Minor-closed: If H G and G 2 C, then H 2 C.
Locally bounded treewidth: tw(G) f (rad(G)) for some f .
Classes C satisfying the above are exactly apex-minor-free.
85
190. Generalizations
We rely on two properties of the graph class C:
Minor-closed: If H G and G 2 C, then H 2 C.
Locally bounded treewidth: tw(G) f (rad(G)) for some f .
Classes C satisfying the above are exactly apex-minor-free.
Corollary: All the results hold on any apex-minor-free class.
85
191. Generalizations
We rely on two properties of the graph class C:
Minor-closed: If H G and G 2 C, then H 2 C.
Locally bounded treewidth: tw(G) f (rad(G)) for some f .
Classes C satisfying the above are exactly apex-minor-free.
Corollary: All the results hold on any apex-minor-free class.
We can generalize to H-minor-free graphs for any H, but we
need to assume that the pattern has bounded maximum
degree.
85
192. Generalizations
We rely on two properties of the graph class C:
Minor-closed: If H G and G 2 C, then H 2 C.
Locally bounded treewidth: tw(G) f (rad(G)) for some f .
Classes C satisfying the above are exactly apex-minor-free.
Corollary: All the results hold on any apex-minor-free class.
We can generalize to H-minor-free graphs for any H, but we
need to assume that the pattern has bounded maximum
degree.
Issue: Apices can “teleport” the pattern almost arbitrarily.
85
193. Generalizations
We rely on two properties of the graph class C:
Minor-closed: If H G and G 2 C, then H 2 C.
Locally bounded treewidth: tw(G) f (rad(G)) for some f .
Classes C satisfying the above are exactly apex-minor-free.
Corollary: All the results hold on any apex-minor-free class.
We can generalize to H-minor-free graphs for any H, but we
need to assume that the pattern has bounded maximum
degree.
Issue: Apices can “teleport” the pattern almost arbitrarily.
Open:
85
194. Generalizations
We rely on two properties of the graph class C:
Minor-closed: If H G and G 2 C, then H 2 C.
Locally bounded treewidth: tw(G) f (rad(G)) for some f .
Classes C satisfying the above are exactly apex-minor-free.
Corollary: All the results hold on any apex-minor-free class.
We can generalize to H-minor-free graphs for any H, but we
need to assume that the pattern has bounded maximum
degree.
Issue: Apices can “teleport” the pattern almost arbitrarily.
Open:
Handling disconnected patterns.
85
195. Generalizations
We rely on two properties of the graph class C:
Minor-closed: If H G and G 2 C, then H 2 C.
Locally bounded treewidth: tw(G) f (rad(G)) for some f .
Classes C satisfying the above are exactly apex-minor-free.
Corollary: All the results hold on any apex-minor-free class.
We can generalize to H-minor-free graphs for any H, but we
need to assume that the pattern has bounded maximum
degree.
Issue: Apices can “teleport” the pattern almost arbitrarily.
Open:
Handling disconnected patterns.
Extending the technique to general H-minor-free graphs.
85
196. Summary of Chapter 4
Contribution: New tool for developing subexp. parameterized
algorithms on topologically-constrained graph classes.
86
197. Summary of Chapter 4
Contribution: New tool for developing subexp. parameterized
algorithms on topologically-constrained graph classes.
Usage: Searching for small patterns with prescribed properties.
86
198. Summary of Chapter 4
Contribution: New tool for developing subexp. parameterized
algorithms on topologically-constrained graph classes.
Usage: Searching for small patterns with prescribed properties.
Seems more robust than bidimensionality.
86
199. Summary of Chapter 4
Contribution: New tool for developing subexp. parameterized
algorithms on topologically-constrained graph classes.
Usage: Searching for small patterns with prescribed properties.
Seems more robust than bidimensionality.
Outlook:
86
200. Summary of Chapter 4
Contribution: New tool for developing subexp. parameterized
algorithms on topologically-constrained graph classes.
Usage: Searching for small patterns with prescribed properties.
Seems more robust than bidimensionality.
Outlook:
A similar tool can be given for graph classes with polynomial
growth (Marx, Pilipczuk; 2016+).
86
201. Summary of Chapter 4
Contribution: New tool for developing subexp. parameterized
algorithms on topologically-constrained graph classes.
Usage: Searching for small patterns with prescribed properties.
Seems more robust than bidimensionality.
Outlook:
A similar tool can be given for graph classes with polynomial
growth (Marx, Pilipczuk; 2016+).
Questions:
Derandomization, disconnected patterns, H-minor-free graphs.
86
202. Conclusions
A robust understanding of why certain problems can be solved
in time 2O(
p
n) etc. on planar graphs and why the square root
is best possible.
87
203. Conclusions
A robust understanding of why certain problems can be solved
in time 2O(
p
n) etc. on planar graphs and why the square root
is best possible.
Going beyond the basic toolbox requires new problem-specific
algorithmic techniques and hardness proofs with tricky gadget
constructions.
87
204. Conclusions
A robust understanding of why certain problems can be solved
in time 2O(
p
n) etc. on planar graphs and why the square root
is best possible.
Going beyond the basic toolbox requires new problem-specific
algorithmic techniques and hardness proofs with tricky gadget
constructions.
The lower bound technology on planar graphs cannot give a
lower bound without a square root factor. Does this mean that
there are matching algorithms for other problems as well?
2O(
p
k)
· nO(1)
time algorithm for Steiner Tree with k
terminals in a planar graph?
. . .
87
205. Conclusions
Chapter 1: Subexponential algorithms using treewidth.
Algorithms: standard treewidth algorithms.
Lower bounds: textbook NP-completeness proofs + ETH.
Chapter 2: Grid minors and bidimensionality.
Algorithms: standard treewidth algorithms + excluded grid
theorem.
Lower bounds: textbook NP-completeness proofs + ETH.
Chapter 3: Finding bounded-treewidth solutions.
Algorithms: the solution can be represented by a graph of
treewidth O(
p
k).
Lower bounds: grid-like W[1]-hardness proofs to rule out
f (k) · no(
p
k)
algorithms.
Chapter 4: Pattern Coverage.
New Tool: A kind of graphic hash functions covering patterns
of treewidth O(
p
k). Much more to be explored here.
88
206. Conclusions
Chapter 1: Subexponential algorithms using treewidth.
Algorithms: standard treewidth algorithms.
Lower bounds: textbook NP-completeness proofs + ETH.
Chapter 2: Grid minors and bidimensionality.
Algorithms: standard treewidth algorithms + excluded grid
theorem.
Lower bounds: textbook NP-completeness proofs + ETH.
Chapter 3: Finding bounded-treewidth solutions.
Algorithms: the solution can be represented by a graph of
treewidth O(
p
k).
Lower bounds: grid-like W[1]-hardness proofs to rule out
f (k) · no(
p
k)
algorithms.
Chapter 4: Pattern Coverage.
New Tool: A kind of graphic hash functions covering patterns
of treewidth O(
p
k). Much more to be explored here.
Thanks for listening!
88