The document describes research into the maximum edge coloring problem, which involves coloring the edges of a graph such that each vertex sees at most two colors. The goal is to maximize the number of colors used. The problem is known to be NP-complete. The authors present a fixed-parameter tractable algorithm that runs in time O*(20k) by reducing the problem into smaller subproblems involving color palettes, vertex covers, and independent sets. They also discuss some open problems regarding improving the running time and determining whether the problem admits a polynomial kernel.
This document provides an overview of mathematical preliminaries including sets, functions, relations, graphs, and proof techniques. It defines sets, set operations, and set representations. It also covers functions, relations, Cartesian products, graphs including walks, paths, cycles, and trees. Finally, it discusses proof techniques like induction and contradiction and provides examples of proofs using these techniques.
The document discusses the problem of finding the longest common subsequence (LCS) between two sequences. It presents a brute force algorithm that has exponential runtime complexity. It then introduces an dynamic programming approach that uses a table c[i,j] to store the length of the LCS between prefixes of the two sequences. The values in the table satisfy a recursive relationship, allowing the LCS to be found in polynomial time complexity. The approach exploits the optimal substructure property of the problem.
Show and tell: A Neural Image caption generatorHojin Yang
ย
The document describes a neural image caption generator that uses a CNN-RNN model. It summarizes the model, which uses a CNN like GoogLeNet to encode an image into a vector, which is then input to an RNN decoder to generate a caption word-by-word. The RNN is trained with cross entropy loss to maximize the probability of the correct caption. It also discusses techniques for variable length captions, sampling methods for inference, and using attention to focus on parts of the image.
MMCF: Multimodal Collaborative Filtering for Automatic Playlist ConitnuationHojin Yang
ย
The slides used for presentation in the 'ecSys challenge workshop 2018'. The challenge is co-organized by Spotify. Our team('hello world!') won the 2nd place.
Strong Edge Coloring for Channel Assignment in Wireless Radio Networksshripadthite
ย
The document describes channel assignment in wireless radio networks as a strong edge coloring problem on graphs. It presents sequential approximation algorithms to minimize the number of channels needed for interference-free communication. It also provides a distributed protocol to maximize the number of communicating transceiver pairs when a fixed number of channels are available. Experimental results show the performance of a greedy distributed algorithm for strong edge coloring on random disk graphs with power-law distributed broadcast radii.
The document contains announcements for students at Brown Mackie College related to various events, deadlines, and opportunities including:
- An invitation for students struggling in classes to visit the Student Success Center for help
- An announcement about a raffle being held by the Education Foundation
- A notice that Keith Grant will be assuming the role of dean during a transition period
- Details about upcoming Ambassador Informational sessions at the Norwood and Woodlawn campuses
- Reminders about textbook purchase and book buyback deadlines in November
- An announcement about a music video contest being held with a deadline of December 20th and awards event on January 23rd
Reaching the last 10% - iNorthumberland - presented by Steve Bluff, OurbroadbandtechUK
ย
Reaching the last 10% - how can Local Government achieve 100% connectivity?
The Solution โ Three Case Studies
iNorthumberland - presented by Steve Bluff, Managing Director, Ourbroadband
For more information: http://www.techuk.org/events/conference/item/826-reaching-the-last-5-how-the-uk-achieves-100-connectivity
All Rights Reserved
The document discusses consolidation trends in the ad tech industry. It notes that VC investment in ad tech has declined as many companies have proven unprofitable. M&A activity is increasing as investors lose patience, with the number of ad tech deals doubling in the first half of 2013 compared to the same period in 2012. The document examines recent ad tech acquisitions and argues that too many small "point solutions" have been funded, leading to unnecessary competition and high costs. It suggests further consolidation is needed among data management platforms and other players.
This document provides an overview of mathematical preliminaries including sets, functions, relations, graphs, and proof techniques. It defines sets, set operations, and set representations. It also covers functions, relations, Cartesian products, graphs including walks, paths, cycles, and trees. Finally, it discusses proof techniques like induction and contradiction and provides examples of proofs using these techniques.
The document discusses the problem of finding the longest common subsequence (LCS) between two sequences. It presents a brute force algorithm that has exponential runtime complexity. It then introduces an dynamic programming approach that uses a table c[i,j] to store the length of the LCS between prefixes of the two sequences. The values in the table satisfy a recursive relationship, allowing the LCS to be found in polynomial time complexity. The approach exploits the optimal substructure property of the problem.
Show and tell: A Neural Image caption generatorHojin Yang
ย
The document describes a neural image caption generator that uses a CNN-RNN model. It summarizes the model, which uses a CNN like GoogLeNet to encode an image into a vector, which is then input to an RNN decoder to generate a caption word-by-word. The RNN is trained with cross entropy loss to maximize the probability of the correct caption. It also discusses techniques for variable length captions, sampling methods for inference, and using attention to focus on parts of the image.
MMCF: Multimodal Collaborative Filtering for Automatic Playlist ConitnuationHojin Yang
ย
The slides used for presentation in the 'ecSys challenge workshop 2018'. The challenge is co-organized by Spotify. Our team('hello world!') won the 2nd place.
Strong Edge Coloring for Channel Assignment in Wireless Radio Networksshripadthite
ย
The document describes channel assignment in wireless radio networks as a strong edge coloring problem on graphs. It presents sequential approximation algorithms to minimize the number of channels needed for interference-free communication. It also provides a distributed protocol to maximize the number of communicating transceiver pairs when a fixed number of channels are available. Experimental results show the performance of a greedy distributed algorithm for strong edge coloring on random disk graphs with power-law distributed broadcast radii.
The document contains announcements for students at Brown Mackie College related to various events, deadlines, and opportunities including:
- An invitation for students struggling in classes to visit the Student Success Center for help
- An announcement about a raffle being held by the Education Foundation
- A notice that Keith Grant will be assuming the role of dean during a transition period
- Details about upcoming Ambassador Informational sessions at the Norwood and Woodlawn campuses
- Reminders about textbook purchase and book buyback deadlines in November
- An announcement about a music video contest being held with a deadline of December 20th and awards event on January 23rd
Reaching the last 10% - iNorthumberland - presented by Steve Bluff, OurbroadbandtechUK
ย
Reaching the last 10% - how can Local Government achieve 100% connectivity?
The Solution โ Three Case Studies
iNorthumberland - presented by Steve Bluff, Managing Director, Ourbroadband
For more information: http://www.techuk.org/events/conference/item/826-reaching-the-last-5-how-the-uk-achieves-100-connectivity
All Rights Reserved
The document discusses consolidation trends in the ad tech industry. It notes that VC investment in ad tech has declined as many companies have proven unprofitable. M&A activity is increasing as investors lose patience, with the number of ad tech deals doubling in the first half of 2013 compared to the same period in 2012. The document examines recent ad tech acquisitions and argues that too many small "point solutions" have been funded, leading to unnecessary competition and high costs. It suggests further consolidation is needed among data management platforms and other players.
This document contains announcements for students at Brown Mackie College Cincinnati including information about tutoring services, raffles, textbook deadlines, shuttle services between campuses, computer lab policies, grade access, singing opportunities for a holiday CD, iPad setup instructions, and a clothing promotion. It provides various updates and reminders to students.
This study guide covers topics in precalculus including:
1) Examples of natural numbers, integers, rational numbers, and real numbers.
2) Composition of functions including finding f(g(x)) and g(f(x)) for given functions f(x) and g(x).
3) Finding inverses of functions and verifying inverses by composition.
4) Operations on complex numbers including plotting, finding modulus, distance, midpoint, addition, subtraction, multiplication, and division.
5) Graphing functions, classifying function types, and stating domains and ranges.
6) Finding limits of functions as x approaches values.
The document expresses gratitude and discusses a caring relationship between two people, referring to them as "U & me". It states that the two people will wish each other well, laugh and cheer together, and smile at one another. The document is signed from "Prof Sumit".
1) The document discusses the creation of a teaser trailer for a sci-fi/horror film called "Pravus" by a group of four students.
2) It provides details on the storyline, genres used, inspiration from other films, branding elements like posters and trailers, and audience feedback.
3) Various software like Final Cut Express and Adobe programs were used to edit footage and design marketing materials for the fictional film.
The document provides guidance for students to analyze a film poster by annotating how it creates meaning for its target audience. It lists techniques used in effective posters, such as use of photo/mise-en-scene to reflect genre, taglines, certification, credits, references to other works, and stars. Students are instructed to choose a poster and write a mini essay evaluating how it appeals to audiences through these techniques rather than just describing the poster. The document emphasizes that posters rely on audiences' previous understanding and use symbols and connotations to convey messages about the film.
Jim McElligott has experience in environmental compliance, structural and flat concrete construction, electrical/electronics design, manufacturing, and product management. He has worked on several construction projects including federal correctional facilities, military bases, and commercial developments. Additionally, he has experience in engineering management, product design, and strategic planning in the electrical/electronics industry. His qualifications include experience estimating, project cost analysis, and regulatory compliance.
A Quest for Subexponential Time Parameterized Algorithms for Planar-k-Path: F...cseiitgn
ย
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(โn) 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(โk) algorithms for problems like k-path, even for some W[1]-hard problems parameterized by k.
Graph coloring is the assignment of colors to the graph vertices and edges in the graph theory. We can
divide the graph coloring in two types. The first is vertex coloring and the second is edge coloring. The
condition which we follow in graph coloring is that the incident vertices/edges have not the same color.
There are some algorithms which solve the problem of graph coloring. Some are offline algorithm and
others are online algorithm. Where offline means the graph is known in advance and the online means that
the edges of the graph are arrive one by one as an input, and We need to color each edge as soon as it is
added to the graph and the main issue is that we want to minimize the number of colors. We cannot change
the color of an edge after colored in an online algorithm. In this paper, we improve the online algorithm
for edge coloring. There is also a theorem which proves that if the maximum degree of a graph is ฮ, then it
is possible to color its edges, in polynomial time, using at most ฮ+ 1 color. The algorithm provided by
Vizing is offline, i.e., it assumes the whole graph is known in advance. In online algorithm edges arrive one
by one in a random permutation. This online algorithm is inspired by a distributed offline algorithm of
Panconesi and Srinivasan, referred as PS algorithm, works on 2-rounds which we extend by reusing colors
online in multiple rounds.
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.
This document discusses graph coloring, which involves assigning colors to the vertices of a graph such that no two adjacent vertices have the same color. It provides examples of problems that can be modeled as graph coloring, such as scheduling committee meetings. The key points covered include defining graph coloring and chromatic number, discussing greedy algorithms and their limitations for graph coloring, and presenting the Welsh-Powell algorithm as an approach to graph coloring.
This document summarizes a presentation on graph coloring. Graph coloring involves assigning colors to the vertices of a graph such that no two adjacent vertices have the same color. It has applications in problems like channel assignment. The document defines key terms like k-coloring, chromatic number, and k-chromatic graphs. It also discusses the NP-complete nature of the graph coloring problem and summarizes basic greedy and Welsh-Powell algorithms for graph coloring.
This document provides an overview of techniques for solving hard computational problems. It discusses the complexity classes P, NP, and NP-complete, and provides examples of NP-complete problems like the travelling salesman problem. The document then discusses heuristic approaches like approximation and randomized algorithms. It also discusses exploiting additional structure in problem inputs and parameterized/exact analysis. Finally, it provides an example of using vertex cover techniques like degree bounds to solve the vertex cover problem in polynomial time for certain cases.
This document discusses graph coloring and its applications. It begins by defining graph coloring as assigning labels or colors to elements of a graph such that no two adjacent elements have the same color. It then provides examples of vertex coloring, edge coloring, and face coloring. The document also discusses the chromatic number and chromatic polynomial. It describes several real-world applications of graph coloring, including frequency assignment in cellular networks.
Dmitry Shabanov โ Improved algorithms for colorings of simple hypergraphs and...Yandex
ย
The famous Lovรกsz Local Lemma was derived in the paper of P. Erdลs and Lovรกsz to prove that any _n_-uniform non-_r_-colorable hypergraph _H_ has maximum edge degree at least
_ฮ(H) ≥ ¼ r_<sup>_nโ1_</sup>.
A long series of papers is devoted to the improvement of this classical result for different classesof uniform hypergraphs.
In our work we deal with colorings of simple hypergraphs, i.e. hypergraphs in which everytwo distinct edges do not share more than one vertex. By using a multipass random recoloringwe show that any simple _n_-uniform non-_r_-colorable hypergraph _H_ has maximum edge degree at least
_ฮ(H) ≥ ั ยท nr_<sup>_nโ1_</sup>
where _c_ > 0 is an absolute constant. We also give some applications of our probabilistic technique, we establish a new lower bound for the Van der Waerden number and extend the main result to the _b_-simple case.
The work of the second author was supported by Russian Foundation of Fundamental Research (grant โย 12-01-00683-a), by the program โLeading Scientific Schoolsโ (grant no. NSh-2964.2014.1) and by the grant of the President of Russian Federation MK-692.2014.1
The document discusses game theory and algorithms used in game-playing programs such as Minimax and Alpha-Beta pruning. It provides examples of how Minimax works to evaluate all possible moves in a game tree and select the optimal move. Alpha-Beta pruning improves on Minimax by avoiding exploring subtrees that cannot affect the outcome, reducing computation time.
The document discusses graph coloring and its applications. It defines graph coloring as assigning colors to vertices of a graph such that no two adjacent vertices have the same color. Applications of graph coloring include frequency assignment in mobile networks, mapping coloration, scheduling problems, Sudoku puzzles, and register allocation. The minimum number of colors needed for a graph is called its chromatic number.
The document presents algorithms for finding the largest induced q-colorable subgraph of a given graph G. It first describes a randomized algorithm that runs in time proportional to enumerating maximal independent sets and a polynomial in n and q. For perfect graphs, where maximum independent sets can be found efficiently, it gives a deterministic algorithm running in similar time. It also shows that the problem does not admit a polynomial kernel when parameterized by the solution size for split and perfect graphs under standard assumptions.
Sparsity Based Super Resolution Using Color Channel ConstraintsHojjat Seyed Mousavi
ย
Sparsity constrained single image super-resolution (SR) has been of much recent interest. A typical approach involves sparsely representing patches in a low-resolution (LR) input image via a dictionary of example LR patches, and then using the coefficients of this representation to generate the high-resolution (HR) output via an analogous HR dictionary. However, most existing sparse representation methods for super resolution focus on the luminance channel information and do not capture interactions between color channels. In this work, we extend sparsity based super-resolution to multiple color channels by taking color information into account. Edge similarities amongst RGB color bands are exploited as cross channel correlation constraints. These additional constraints lead to a new optimization problem which is not easily solvable; however, a tractable solution is proposed to solve it efficiently. Moreover, to fully exploit the complementary information among color channels, a dictionary learning method is also proposed specifically to learn color dictionaries that encourage edge similarities. Merits of the proposed method over state of the art are demonstrated both visually and quantitatively using image quality metrics.
This document summarizes 9 problems from the ACM ICPC 2013-2014 Northeastern European Regional Contest. It provides an overview of the key algorithms and approaches for each problem, describing them concisely in 1-3 sentences per problem. The problems cover a range of algorithmic topics including exhaustive search, dynamic programming, graph algorithms, geometry, and more.
Analysis and design of algorithms part 4Deepak John
ย
Complexity Theory - Introduction. P and NP. NP-Complete problems. Approximation algorithms. Bin packing, Graph coloring. Traveling salesperson Problem.
Hidden Markov Model in Natural Language Processingsachinmaskeen211
ย
This document discusses hidden Markov models and the forward-backward algorithm. It introduces marginalization and conditionalization concepts using sales data examples. It then explains how these concepts apply to a weather prediction example modeled as a hidden Markov model. The document discusses computing alpha and beta values using the forward-backward algorithm to find the most likely hidden state sequence. It also discusses how hidden Markov models can be used for part-of-speech tagging of text.
This document contains announcements for students at Brown Mackie College Cincinnati including information about tutoring services, raffles, textbook deadlines, shuttle services between campuses, computer lab policies, grade access, singing opportunities for a holiday CD, iPad setup instructions, and a clothing promotion. It provides various updates and reminders to students.
This study guide covers topics in precalculus including:
1) Examples of natural numbers, integers, rational numbers, and real numbers.
2) Composition of functions including finding f(g(x)) and g(f(x)) for given functions f(x) and g(x).
3) Finding inverses of functions and verifying inverses by composition.
4) Operations on complex numbers including plotting, finding modulus, distance, midpoint, addition, subtraction, multiplication, and division.
5) Graphing functions, classifying function types, and stating domains and ranges.
6) Finding limits of functions as x approaches values.
The document expresses gratitude and discusses a caring relationship between two people, referring to them as "U & me". It states that the two people will wish each other well, laugh and cheer together, and smile at one another. The document is signed from "Prof Sumit".
1) The document discusses the creation of a teaser trailer for a sci-fi/horror film called "Pravus" by a group of four students.
2) It provides details on the storyline, genres used, inspiration from other films, branding elements like posters and trailers, and audience feedback.
3) Various software like Final Cut Express and Adobe programs were used to edit footage and design marketing materials for the fictional film.
The document provides guidance for students to analyze a film poster by annotating how it creates meaning for its target audience. It lists techniques used in effective posters, such as use of photo/mise-en-scene to reflect genre, taglines, certification, credits, references to other works, and stars. Students are instructed to choose a poster and write a mini essay evaluating how it appeals to audiences through these techniques rather than just describing the poster. The document emphasizes that posters rely on audiences' previous understanding and use symbols and connotations to convey messages about the film.
Jim McElligott has experience in environmental compliance, structural and flat concrete construction, electrical/electronics design, manufacturing, and product management. He has worked on several construction projects including federal correctional facilities, military bases, and commercial developments. Additionally, he has experience in engineering management, product design, and strategic planning in the electrical/electronics industry. His qualifications include experience estimating, project cost analysis, and regulatory compliance.
A Quest for Subexponential Time Parameterized Algorithms for Planar-k-Path: F...cseiitgn
ย
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(โn) 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(โk) algorithms for problems like k-path, even for some W[1]-hard problems parameterized by k.
Graph coloring is the assignment of colors to the graph vertices and edges in the graph theory. We can
divide the graph coloring in two types. The first is vertex coloring and the second is edge coloring. The
condition which we follow in graph coloring is that the incident vertices/edges have not the same color.
There are some algorithms which solve the problem of graph coloring. Some are offline algorithm and
others are online algorithm. Where offline means the graph is known in advance and the online means that
the edges of the graph are arrive one by one as an input, and We need to color each edge as soon as it is
added to the graph and the main issue is that we want to minimize the number of colors. We cannot change
the color of an edge after colored in an online algorithm. In this paper, we improve the online algorithm
for edge coloring. There is also a theorem which proves that if the maximum degree of a graph is ฮ, then it
is possible to color its edges, in polynomial time, using at most ฮ+ 1 color. The algorithm provided by
Vizing is offline, i.e., it assumes the whole graph is known in advance. In online algorithm edges arrive one
by one in a random permutation. This online algorithm is inspired by a distributed offline algorithm of
Panconesi and Srinivasan, referred as PS algorithm, works on 2-rounds which we extend by reusing colors
online in multiple rounds.
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.
This document discusses graph coloring, which involves assigning colors to the vertices of a graph such that no two adjacent vertices have the same color. It provides examples of problems that can be modeled as graph coloring, such as scheduling committee meetings. The key points covered include defining graph coloring and chromatic number, discussing greedy algorithms and their limitations for graph coloring, and presenting the Welsh-Powell algorithm as an approach to graph coloring.
This document summarizes a presentation on graph coloring. Graph coloring involves assigning colors to the vertices of a graph such that no two adjacent vertices have the same color. It has applications in problems like channel assignment. The document defines key terms like k-coloring, chromatic number, and k-chromatic graphs. It also discusses the NP-complete nature of the graph coloring problem and summarizes basic greedy and Welsh-Powell algorithms for graph coloring.
This document provides an overview of techniques for solving hard computational problems. It discusses the complexity classes P, NP, and NP-complete, and provides examples of NP-complete problems like the travelling salesman problem. The document then discusses heuristic approaches like approximation and randomized algorithms. It also discusses exploiting additional structure in problem inputs and parameterized/exact analysis. Finally, it provides an example of using vertex cover techniques like degree bounds to solve the vertex cover problem in polynomial time for certain cases.
This document discusses graph coloring and its applications. It begins by defining graph coloring as assigning labels or colors to elements of a graph such that no two adjacent elements have the same color. It then provides examples of vertex coloring, edge coloring, and face coloring. The document also discusses the chromatic number and chromatic polynomial. It describes several real-world applications of graph coloring, including frequency assignment in cellular networks.
Dmitry Shabanov โ Improved algorithms for colorings of simple hypergraphs and...Yandex
ย
The famous Lovรกsz Local Lemma was derived in the paper of P. Erdลs and Lovรกsz to prove that any _n_-uniform non-_r_-colorable hypergraph _H_ has maximum edge degree at least
_ฮ(H) ≥ ¼ r_<sup>_nโ1_</sup>.
A long series of papers is devoted to the improvement of this classical result for different classesof uniform hypergraphs.
In our work we deal with colorings of simple hypergraphs, i.e. hypergraphs in which everytwo distinct edges do not share more than one vertex. By using a multipass random recoloringwe show that any simple _n_-uniform non-_r_-colorable hypergraph _H_ has maximum edge degree at least
_ฮ(H) ≥ ั ยท nr_<sup>_nโ1_</sup>
where _c_ > 0 is an absolute constant. We also give some applications of our probabilistic technique, we establish a new lower bound for the Van der Waerden number and extend the main result to the _b_-simple case.
The work of the second author was supported by Russian Foundation of Fundamental Research (grant โย 12-01-00683-a), by the program โLeading Scientific Schoolsโ (grant no. NSh-2964.2014.1) and by the grant of the President of Russian Federation MK-692.2014.1
The document discusses game theory and algorithms used in game-playing programs such as Minimax and Alpha-Beta pruning. It provides examples of how Minimax works to evaluate all possible moves in a game tree and select the optimal move. Alpha-Beta pruning improves on Minimax by avoiding exploring subtrees that cannot affect the outcome, reducing computation time.
The document discusses graph coloring and its applications. It defines graph coloring as assigning colors to vertices of a graph such that no two adjacent vertices have the same color. Applications of graph coloring include frequency assignment in mobile networks, mapping coloration, scheduling problems, Sudoku puzzles, and register allocation. The minimum number of colors needed for a graph is called its chromatic number.
The document presents algorithms for finding the largest induced q-colorable subgraph of a given graph G. It first describes a randomized algorithm that runs in time proportional to enumerating maximal independent sets and a polynomial in n and q. For perfect graphs, where maximum independent sets can be found efficiently, it gives a deterministic algorithm running in similar time. It also shows that the problem does not admit a polynomial kernel when parameterized by the solution size for split and perfect graphs under standard assumptions.
Sparsity Based Super Resolution Using Color Channel ConstraintsHojjat Seyed Mousavi
ย
Sparsity constrained single image super-resolution (SR) has been of much recent interest. A typical approach involves sparsely representing patches in a low-resolution (LR) input image via a dictionary of example LR patches, and then using the coefficients of this representation to generate the high-resolution (HR) output via an analogous HR dictionary. However, most existing sparse representation methods for super resolution focus on the luminance channel information and do not capture interactions between color channels. In this work, we extend sparsity based super-resolution to multiple color channels by taking color information into account. Edge similarities amongst RGB color bands are exploited as cross channel correlation constraints. These additional constraints lead to a new optimization problem which is not easily solvable; however, a tractable solution is proposed to solve it efficiently. Moreover, to fully exploit the complementary information among color channels, a dictionary learning method is also proposed specifically to learn color dictionaries that encourage edge similarities. Merits of the proposed method over state of the art are demonstrated both visually and quantitatively using image quality metrics.
This document summarizes 9 problems from the ACM ICPC 2013-2014 Northeastern European Regional Contest. It provides an overview of the key algorithms and approaches for each problem, describing them concisely in 1-3 sentences per problem. The problems cover a range of algorithmic topics including exhaustive search, dynamic programming, graph algorithms, geometry, and more.
Analysis and design of algorithms part 4Deepak John
ย
Complexity Theory - Introduction. P and NP. NP-Complete problems. Approximation algorithms. Bin packing, Graph coloring. Traveling salesperson Problem.
Hidden Markov Model in Natural Language Processingsachinmaskeen211
ย
This document discusses hidden Markov models and the forward-backward algorithm. It introduces marginalization and conditionalization concepts using sales data examples. It then explains how these concepts apply to a weather prediction example modeled as a hidden Markov model. The document discusses computing alpha and beta values using the forward-backward algorithm to find the most likely hidden state sequence. It also discusses how hidden Markov models can be used for part-of-speech tagging of text.
The document discusses the NP-hard Max Cut problem and provides a reduction from the NP-hard NAE-3-SAT problem to Max Cut to prove that Max Cut is also NP-hard. The reduction works by mapping clauses in a NAE-3-SAT instance to a graph instance of Max Cut, such that a solution to one problem can be translated to a solution for the other problem in polynomial time. This shows that any polynomial time algorithm for Max Cut could also be used to solve NAE-3-SAT in polynomial time. The document then provides a simple randomized approximation algorithm for Max Cut that runs in linear time.
The document summarizes a lecture on blending, compositing, and anti-aliasing in computer graphics. It discusses how colors are combined during rendering using blending operations, and how compositing operates on entire images rather than individual pixels. Porter-Duff models for digital image compositing are explained, along with how they relate to OpenGL blending functions.
The document discusses connected dominating sets and short cycles. It begins by explaining that excluding longer cycles makes related problems easier to solve. Specifically, it shows that on graphs with girth at least five, high degree vertices must be in any minimum dominating set. However, this does not hold for connected dominating sets, since connectivity must also be maintained. It then describes how to obtain fixed-parameter tractable algorithms for connected dominating set problems by guessing the minimum dominating set and extending it. It also shows that these problems do not admit polynomial kernels by providing a reduction from Fair Connected Colors, which is W-hard.
The document discusses Kolmogorov complexity as a framework for understanding randomness and probability. It covers several key topics:
1) Kolmogorov complexity is defined as the length of the shortest program that produces a given string, and can be used to formalize the notion of randomness as incompressibility.
2) An algorithmic approach to probability theory is presented where a random sequence is defined as one whose prefixes have high Kolmogorov complexity. This aligns with the classical notion that random sequences satisfy laws of large numbers.
3) The complexity of sampling truly random strings from a device is discussed, and how the algorithmic and classical views of randomness relate under certain assumptions.
4) Examples are given
Similar to An FPT Algorithm for Maximum Edge Coloring (20)
Efficient algorithms for hard problems on structured electoratesNeeldhara Misra
ย
This talk explores possibilities for exploiting structure in voting profiles to obtain efficient algorithms for problems that are computationally intractable in general.
On the Parameterized Complexity of Party NominationsNeeldhara Misra
ย
Consider a fixed voting rule R. In the Possible President problem, we are given an election where the candidates are partitioned into parties, and the problem is to determine if, given a party P, it is possible for every party to nominate a candidate such that the nominee from P is a winner of the election that is obtained by restricting the votes to the nominated candidates. In the Necessary President problem, we would like to find a nominee who wins no matter who else is nominated. In this talk, we explore the complexity of these problems, which can be thought of as the two natural extremes of the party nomination problem, with an emphasis on a parameterized perspective and algorithms on structured profiles.
We consider a natural variant of the well-known Feedback Vertex Set problem, namely the problem of deleting a small subset of vertices or edges to a full binary tree. This version of the problem is motivated by real-world scenarios that are best modeled by full binary trees. We establish that both versions of the problem are NP-hard, which stands in contrast to the fact that deleting edges to obtain a forest or a tree is equivalent to the problem of finding a minimum cost spanning tree, which can be solved in polynomial time. We also establish that both problems are FPT by the standard parameter.
Elicitation for Preferences Single Peaked on Trees Neeldhara Misra
ย
This talk will focus on the problem of preference elicitation, where the goal is to understand the preferences of agents (which we model by total orders) by querying them about their pairwise preferences. We will survey known results, which have studied the problem both on general domains and structured ones, such as the domain of single-peaked preferences. As one might expect, structured domains admit a lower query complexity. We will consider domains that are single peaked over trees, which generalize the notion of single-peakedness.
This document discusses reasons for pursuing research in computer science, including the challenges, joys, and mindset required. It addresses circumstances around research, fascination with problems, eureka moments versus dull periods, the importance of persistence and breadth versus depth. It provides advice on coming to terms with limitations, the social aspects of research, balancing theory and practice, and having other interests besides work. Overall, the document presents research as rewarding but requiring hard work.
The document discusses matchings in graphs and the Erdos-Ko-Rado (EKR) theorem. It introduces Baranyai partitions, which is a decomposition of the edges of a complete bipartite graph K2n into (2n-1) perfect matchings. Considering cyclic permutations of the edges within each perfect matching partition provides a way to set up "Katona-like local environments" to prove bounds on intersecting families of matchings, in analogy to Katona's proof technique for intersecting families of sets.
The document discusses connected separators and 2-connected separators in graphs. It presents the treewidth reduction theorem, which shows that the 2-connected separator problem can be solved by finding an equivalent instance on a graph of small treewidth. It also discusses properties of 2-connected Steiner trees, including that the non-terminal vertices induce a forest, and presents an algorithm that guesses and maps the structure of the 2-connected Steiner tree.
The document describes a generic algorithm for the F-deletion problem, where the goal is to remove at most k vertices from a graph such that the remaining graph does not contain graphs from F as minors. It shows that when F contains only planar graphs, the algorithm provides a constant-factor approximation. It analyzes special cases where the algorithm works with different constants, such as when the graph minus the solution is independent, a matching, or acyclic. It then discusses how the algorithm extends to more general graphs by exploiting that the graph minus solution must have bounded treewidth when F contains planar graphs.
A Kernel for Planar F-deletion: The Connected CaseNeeldhara Misra
ย
The document discusses polynomial kernels for planar F-deletion problems. It presents an algorithm that works by guessing the existence of protrusions and either reducing them or inferring irrelevant edges. If every guess finds an irrelevant edge, those edges can safely be removed from the graph. The algorithm aims to obtain a graph with constant treewidth by repeatedly guessing and reducing protrusions or deleting irrelevant edges. This leads to a polynomial kernel for certain cases of planar F-deletion.
Kernels for Planar F-Deletion (Restricted Variants)Neeldhara Misra
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The document discusses kernelization for the F-deletion problem, where graphs in F are connected and at least one is planar. It is shown that the planar F-deletion problem admits a polynomial kernel whenever F contains a planar graph called the "onion" graph. Several other positive and negative results are also presented, including that planar F-deletion admits an approximation algorithm and a polynomial kernel on claw-free graphs. The document concludes by outlining the ingredients for showing that planar F-deletion admits a polynomial kernel.
Efficient Simplification: The (im)possibilitiesNeeldhara Misra
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The document discusses techniques for simplifying problems, including kernelization procedures. It proposes definitions for what constitutes a good simplification procedure and kernelization procedure. A kernelization procedure takes an input of size n and parameter k and maps it to an equivalent instance of size only g(k) in polynomial time, where g is some computable function. This implies the problem is fixed-parameter tractable. The document also discusses how some NP-complete problems may still admit efficient simplification procedures when restricted to instances with certain properties, like bounded degree graphs.
This document discusses the kernelization complexity of finding colorful motifs in graphs. It introduces colorful motifs as a problem with applications in bioinformatics. It shows the problem is NP-complete even on very simple graph classes from the kernelization perspective. It presents an observation that leads to many polynomial kernels in a special case of comb graphs, but notes this does not generalize. It also provides NP-hardness results and observations ruling out polynomial kernels for more general graph classes and problems.
The document discusses q-expansions and their applications to problems like vertex cover and feedback vertex set. It introduces the q-expansion lemma, which states that if the neighborhood of a set S is at least q times the size of S, then there exist q matchings saturating S that are disjoint in the neighborhood. This lemma is then used to obtain polynomial kernelizations for problems like vertex cover and feedback vertex set by finding a high-degree vertex v and using q-expansions to find a small hitting set that does not contain v. The technique can be generalized to finding solutions for graphs excluding a fixed minor H by using q-expansions to find a small set avoiding a high degree vertex v.
The document discusses kernelization procedures for parameterized problems. It begins by defining kernelization as a polynomial-time preprocessing function that maps an input instance to an equivalent, compressed instance whose size depends only on the parameter. It then proves that a problem admits a kernel (can be kernelized) if and only if it is fixed-parameter tractable. Specifically, a kernel implies an FPT algorithm, and an FPT runtime implies the existence of a kernel. The document advocates for polynomial-sized kernels as the most efficient type of kernelization.
How Barcodes Can Be Leveraged Within Odoo 17Celine George
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In this presentation, we will explore how barcodes can be leveraged within Odoo 17 to streamline our manufacturing processes. We will cover the configuration steps, how to utilize barcodes in different manufacturing scenarios, and the overall benefits of implementing this technology.
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
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Ivรกn Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
This presentation was provided by Racquel Jemison, Ph.D., Christina MacLaughlin, Ph.D., and Paulomi Majumder. Ph.D., all of the American Chemical Society, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.
Elevate Your Nonprofit's Online Presence_ A Guide to Effective SEO Strategies...TechSoup
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Whether you're new to SEO or looking to refine your existing strategies, this webinar will provide you with actionable insights and practical tips to elevate your nonprofit's online presence.
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
ย
(๐๐๐ ๐๐๐) (๐๐๐ฌ๐ฌ๐จ๐ง ๐)-๐๐ซ๐๐ฅ๐ข๐ฆ๐ฌ
๐๐ข๐ฌ๐๐ฎ๐ฌ๐ฌ ๐ญ๐ก๐ ๐๐๐ ๐๐ฎ๐ซ๐ซ๐ข๐๐ฎ๐ฅ๐ฎ๐ฆ ๐ข๐ง ๐ญ๐ก๐ ๐๐ก๐ข๐ฅ๐ข๐ฉ๐ฉ๐ข๐ง๐๐ฌ:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
๐๐ฑ๐ฉ๐ฅ๐๐ข๐ง ๐ญ๐ก๐ ๐๐๐ญ๐ฎ๐ซ๐ ๐๐ง๐ ๐๐๐จ๐ฉ๐ ๐จ๐ ๐๐ง ๐๐ง๐ญ๐ซ๐๐ฉ๐ซ๐๐ง๐๐ฎ๐ซ:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
20. Motivation
In a network, every system has two interface cards.
The goal is to assign frequency channels so that:
..1 No system is assigned more than two channels.
..2 The number of channels used overall is maximized.
21. Motivation
In a graph, every system has two interface cards.
The goal is to assign frequency channels so that:
..1 No system is assigned more than two channels.
..2 The number of channels used overall is maximized.
22. Motivation
In a graph, every vertex has two interface cards.
The goal is to assign frequency channels so that:
..1 No system is assigned more than two channels.
..2 The number of channels used overall is maximized.
23. Motivation
In a graph, every vertex has two interface cards.
The goal is to assign frequency channels so that:
..1 No vertex sees more than two colors.
..2 The number of channels used overall is maximized.
24. Motivation
In a graph, every vertex has two interface cards.
The goal is to assign frequency channels so that:
..1 No vertex sees more than two colors.
..2 The number of colors used overall is maximimized.
25. Past Work
Max Edge coloring is known to be NP-Complete and also APX-Hard (Adamaszek
and Popa, 2010)
26. Past Work
Max Edge coloring is known to be NP-Complete and also APX-Hard (Adamaszek
and Popa, 2010)
A 2-approximation algorithm is known on general graphs (Feng, Zhang and Wang,
2009)
27. Past Work
Max Edge coloring is known to be NP-Complete and also APX-Hard (Adamaszek
and Popa, 2010)
A 2-approximation algorithm is known on general graphs (Feng, Zhang and Wang,
2009)
The problem is shown to have a polynomial time algorithm for complete graphs
and trees (Feng, Zhang and Wang, 2009)
28. Past Work
Max Edge coloring is known to be NP-Complete and also APX-Hard (Adamaszek
and Popa, 2010)
A 2-approximation algorithm is known on general graphs (Feng, Zhang and Wang,
2009)
The problem is shown to have a polynomial time algorithm for complete graphs
and trees (Feng, Zhang and Wang, 2009)
There exists a 5
3 -approximation algorithm for graphs with perefect matching
(Adamaszek and Popa, 2010)
39. The Maximum Edge Coloring Problem (Parameterized)
Input: A graph G and an integer k.
Question: Can the edges of G be colored with k colors so that no vertex
sees more than two colors?
Parameter: k
40. The Maximum Edge Coloring Problem (Parameterized)
Input: A graph G and an integer k.
Question: Can the edges of G be colored with k colors so that no vertex
sees more than two colors?
Parameter: k
42. A parameterized problem is denoted by a pair (Q, k) โ ฮฃโ ร N.
The ๏ฌrst component Q is a classical language, and the number k is called the
parameter.
43. A parameterized problem is denoted by a pair (Q, k) โ ฮฃโ ร N.
The ๏ฌrst component Q is a classical language, and the number k is called the
parameter.
Such a problem is ๏ฌxedโparameter tractable or FPT if there exists an algorithm
that decides it in time O(f(k)nO(1)) on instances of size n.
70. To realize a palette assignment, we must assign colors so that:
71. To realize a palette assignment, we must assign colors so that:
..1 Every edge respects the palette.
..
VertexCover
.
IndependentSet
72. To realize a palette assignment, we must assign colors so that:
..1 Every edge respects the palette.
..2 Every palette is satisi๏ฌed.
.......
VertexCover
.
IndependentSet
96. Whenever a color in X is assigned to an edge, mark it as used.
Branch only over unused colors.
97. Whenever a color in X is assigned to an edge, mark it as used.
Branch only over unused colors.
Once all colors in X are used, assign colors arbitrarily.
112. As it turns out, there are only two kinds of lists:
113. As it turns out, there are only two kinds of lists:
..1 Those with constant size.
..2 Those with a common color.
114. As it turns out, there are only two kinds of lists:
..1 Those with constant size.
Continue to branch.
..2 Those with a common color.
115. As it turns out, there are only two kinds of lists:
..1 Those with constant size.
Continue to branch.
..2 Those with a common color.
Reduces to a maximum matching problem.
122. Other Results
..1 We show an explicit exponential kernel by the application of some simple
reduction rules.
123. Other Results
..1 We show an explicit exponential kernel by the application of some simple
reduction rules.
..2 We also show NP-hardness and polynomial kernels for restricted graph
classes (constant maximum degree, and C4-free graphs).
124. Other Results
..1 We show an explicit exponential kernel by the application of some simple
reduction rules.
..2 We also show NP-hardness and polynomial kernels for restricted graph
classes (constant maximum degree, and C4-free graphs).
..3 We consider the dual parameter 1 and show a polynomial kernel in this
setting.
1
Can we color with at least (n โ k) colors?
126. Several Open Problems!
..1 Can the algorithm be improved to a running time of O(ck) for some
constant c?
127. Several Open Problems!
..1 Can the algorithm be improved to a running time of O(ck) for some
constant c?
..2 Does the problem admit a polynomial kernel?
128. Several Open Problems!
..1 Can the algorithm be improved to a running time of O(ck) for some
constant c?
..2 Does the problem admit a polynomial kernel?
..3 A natural extension would be the above-guarantee version: can we color
with at least (t + k) colors, where t is the size of a maximum matching?
129. Several Open Problems!
..1 Can the algorithm be improved to a running time of O(ck) for some
constant c?
..2 Does the problem admit a polynomial kernel?
..3 A natural extension would be the above-guarantee version: can we color
with at least (t + k) colors, where t is the size of a maximum matching?
..4 Is there an explicit FPT algorithm for the dual parameter?