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.
2. Motivation for Parameterized
Complexity
• Inputs in practice have nice “structure” (read: parameters
beyond just the input size) that can be exploited by
algorithms.
1
3. Motivation for Parameterized
Complexity
• Inputs in practice have nice “structure” (read: parameters
beyond just the input size) that can be exploited by
algorithms.
• This could possibly explain why NP-hard problems (say
SAT) are sometimes solvable well “in practice”.
(Ongoing Program)
1
4. Motivation for Parameterized
Complexity
• Inputs in practice have nice “structure” (read: parameters
beyond just the input size) that can be exploited by
algorithms.
• This could possibly explain why NP-hard problems (say
SAT) are sometimes solvable well “in practice”.
(Ongoing Program)
• Type Checking of ML programs can be done well in practice
though the problem is EXP-hard;
1
5. Motivation for Parameterized
Complexity
• Inputs in practice have nice “structure” (read: parameters
beyond just the input size) that can be exploited by
algorithms.
• This could possibly explain why NP-hard problems (say
SAT) are sometimes solvable well “in practice”.
(Ongoing Program)
• Type Checking of ML programs can be done well in practice
though the problem is EXP-hard; One explanation: O(2kn)
time where k is the alternation depth of the program.
1
6. Motivation for Parameterized
Complexity
• Inputs in practice have nice “structure” (read: parameters
beyond just the input size) that can be exploited by
algorithms.
• This could possibly explain why NP-hard problems (say
SAT) are sometimes solvable well “in practice”.
(Ongoing Program)
• Type Checking of ML programs can be done well in practice
though the problem is EXP-hard; One explanation: O(2kn)
time where k is the alternation depth of the program.
• However, much of early work on PC (vertex cover, feedback
vertex set – undirected and directed, longest path/cycle,
steiner tree, ..) used the solution size as the parameter.
1
7. Motivation for Parameterized
Complexity
• Inputs in practice have nice “structure” (read: parameters
beyond just the input size) that can be exploited by
algorithms.
• This could possibly explain why NP-hard problems (say
SAT) are sometimes solvable well “in practice”.
(Ongoing Program)
• Type Checking of ML programs can be done well in practice
though the problem is EXP-hard; One explanation: O(2kn)
time where k is the alternation depth of the program.
• However, much of early work on PC (vertex cover, feedback
vertex set – undirected and directed, longest path/cycle,
steiner tree, ..) used the solution size as the parameter.
• One possible exception – Treewidth
1
9. This talk
• Some results and open problems on Alternate
Parameterizations
2
10. This talk
• Some results and open problems on Alternate
Parameterizations (Not a comprehensive survey)
2
11. This talk
• Some results and open problems on Alternate
Parameterizations (Not a comprehensive survey)
• Parameters are some functions of input or output, but not
JUST output size.
2
12. Alternate Parameterizations
or
Ecology of Parameters
Venkatesh Raman
The Institute of Mathematical Sciences, Chennai, India.
NMI Workshop on Cryptography and Complexity,
IIT Gandhinagar, November 5, 2016
3
13. Outline
1 Input Parameterizations
Forest + k vertices
Bipartite + k vertices?
Another way to measure closeness to bipartite graphs
Other Highlights/Results/Open Problems
Parameter Ecology
Input parameterizations within P
2 Output Parameterizations
Motivation
Above guarantee parameterization
Below Guarantee Parameterization
Above guarantee parameterization within P
4
15. Forest + k vertices
• In trees, and more generally, in forests, VERTEX COVER,
FEEDBACK VERTEX SET, DOMINATING SET, COLORING are all
solvable in polynomial time.
5
16. Forest + k vertices
• In trees, and more generally, in forests, VERTEX COVER,
FEEDBACK VERTEX SET, DOMINATING SET, COLORING are all
solvable in polynomial time.
• What about in a graph that is a forest + k vertices? Are
these problems FPT when parameterized by k?
5
17. Forest + k vertices
• In trees, and more generally, in forests, VERTEX COVER,
FEEDBACK VERTEX SET, DOMINATING SET, COLORING are all
solvable in polynomial time.
• What about in a graph that is a forest + k vertices? Are
these problems FPT when parameterized by k?
• Note that k is the size of feedback vertex set,
5
18. Forest + k vertices
• In trees, and more generally, in forests, VERTEX COVER,
FEEDBACK VERTEX SET, DOMINATING SET, COLORING are all
solvable in polynomial time.
• What about in a graph that is a forest + k vertices? Are
these problems FPT when parameterized by k?
• Note that k is the size of feedback vertex set, and we are
parameterizing these problems by the size of feedback
vertex set of the input graph.
5
19. VC, MIS, FVS, DomSet, Coloring,
Clique in Forest + k vertices
Definition
• Input: A graph G and a FVS S of G of size k, an integer
• Parameter: k
• Question: Does G have a VC/DomSet/FVS/Chromatic
number of size at most or IS of size at least ? (Or
essentially solve the optimization problems.)
6
20. VC, MIS, FVS, DomSet, Coloring,
Clique in Forest + k vertices
Definition
• Input: A graph G and a FVS S of G of size k, an integer
• Parameter: k
• Question: Does G have a VC/DomSet/FVS/Chromatic
number of size at most or IS of size at least ? (Or
essentially solve the optimization problems.)
Theorem: FVS parameterized by FVS is FPT.
6
21. VC, MIS, FVS, DomSet, Coloring,
Clique in Forest + k vertices
Definition
• Input: A graph G and a FVS S of G of size k, an integer
• Parameter: k
• Question: Does G have a VC/DomSet/FVS/Chromatic
number of size at most or IS of size at least ? (Or
essentially solve the optimization problems.)
Theorem: FVS parameterized by FVS is FPT.
Proof:
6
22. VC, MIS, FVS, DomSet, Coloring,
Clique in Forest + k vertices
Definition
• Input: A graph G and a FVS S of G of size k, an integer
• Parameter: k
• Question: Does G have a VC/DomSet/FVS/Chromatic
number of size at most or IS of size at least ? (Or
essentially solve the optimization problems.)
Theorem: FVS parameterized by FVS is FPT.
Proof:
• FVS If ≥ k, then YES,
6
23. VC, MIS, FVS, DomSet, Coloring,
Clique in Forest + k vertices
Definition
• Input: A graph G and a FVS S of G of size k, an integer
• Parameter: k
• Question: Does G have a VC/DomSet/FVS/Chromatic
number of size at most or IS of size at least ? (Or
essentially solve the optimization problems.)
Theorem: FVS parameterized by FVS is FPT.
Proof:
• FVS If ≥ k, then YES,
else ≤ k, apply the O∗(5 ) algorithm, that is O∗(5k).
6
24. Vertex Cover in Forest + k vertices
S
Y
As Red vertices from S are not picked into solution, the
Red vertices from V S are forced to be picked into solution.
7
25. Vertex Cover in Forest + k vertices
S
Y
As Red vertices from S are not picked into solution, the
Red vertices from V S are forced to be picked into solution.
7
26. Vertex Cover in Forest + k vertices
S
Y
As Red vertices from S are not picked into solution, the
Red vertices from V S are forced to be picked into solution.
• Guess the intersection Y of solution with S, delete Y,
7
27. Vertex Cover in Forest + k vertices
S
Y
As Red vertices from S are not picked into solution, the
Red vertices from V S are forced to be picked into solution.
• Guess the intersection Y of solution with S, delete Y,
• Pick N(S Y) ∩ (V S) into the solution, delete them
7
28. Vertex Cover in Forest + k vertices
S
Y
As Red vertices from S are not picked into solution, the
Red vertices from V S are forced to be picked into solution.
• Guess the intersection Y of solution with S, delete Y,
• Pick N(S Y) ∩ (V S) into the solution, delete them
• Solve the (optimum) VC problem in the remaining forest.
7
29. Vertex Cover in Forest + k vertices
S
Y
As Red vertices from S are not picked into solution, the
Red vertices from V S are forced to be picked into solution.
• Guess the intersection Y of solution with S, delete Y,
• Pick N(S Y) ∩ (V S) into the solution, delete them
• Solve the (optimum) VC problem in the remaining forest.
7
30. VC in F + k vertices where F is a class
where VC is in P
8
31. VC in F + k vertices where F is a class
where VC is in P
• Guess the intersection Y of solution with S, delete Y,
• Pick N(S Y) ∩ (V S) into the solution, delete them
• Solve the (optimum) VC problem in the remaining graph in
F.
S
Y
As Red vertices from S are not picked into solution, the
Red vertices from V S are forced to be picked into solution.
8
33. Dominating Set, Chromatic Number in
Forest + k vertices
Theorem They are FPT as G has treewidth at most k + 1, and
DomSet, Coloring are FPT parameterized by treewidth
9
34. Dominating Set, Chromatic Number in
Forest + k vertices
Theorem They are FPT as G has treewidth at most k + 1, and
DomSet, Coloring are FPT parameterized by treewidth
• Forest + k edges?
9
35. Dominating Set, Chromatic Number in
Forest + k vertices
Theorem They are FPT as G has treewidth at most k + 1, and
DomSet, Coloring are FPT parameterized by treewidth
• Forest + k edges? Similar results go through as FAS of size
at most k implies FVS of size at most 2k.
9
37. Kernelization for VC in F+k graphs
where F is a class where VC is in P
Kernelization question tends to be much harder.
10
38. Kernelization for VC in F+k graphs
where F is a class where VC is in P
Kernelization question tends to be much harder.
VERTEX COVER in F + k graph
• has an O(k3) kernel when parameterized by FVS (FJR
2013).
• has an O(k5) vertex kernel if F is the class of degree at
most 2 graphs (MRS IPEC 2015);
• has an O(k12) vertex kernel if F is the class of psuedoforests
(each component has at most one cycle) (FS IPEC 2016);
• has an O(kd) vertex kernel if F is the class of bounded
cluster graphs (each component is a cluster of size at most
d); (MRS IPEC 2015)
• but has no polynomial kernel (under complexity
conditions) if F is a collection of all graphs where each
component is a (unbounded size) clique. (FJR 2013)
10
39. Kernelization for VC in F+k graphs
where F is a class where VC is in P
Kernelization question tends to be much harder.
VERTEX COVER in F + k graph
• has an O(k3) kernel when parameterized by FVS (FJR
2013).
• has an O(k5) vertex kernel if F is the class of degree at
most 2 graphs (MRS IPEC 2015);
• has an O(k12) vertex kernel if F is the class of psuedoforests
(each component has at most one cycle) (FS IPEC 2016);
• has an O(kd) vertex kernel if F is the class of bounded
cluster graphs (each component is a cluster of size at most
d); (MRS IPEC 2015)
• but has no polynomial kernel (under complexity
conditions) if F is a collection of all graphs where each
component is a (unbounded size) clique. (FJR 2013)
• A dichotomy result for kernels here? OPEN
10
40. Parameterized by the size of Odd Cycle
Transversal (I.e. Bipartite + k vertices)
11
41. Parameterized by the size of Odd Cycle
Transversal (I.e. Bipartite + k vertices)
11
42. Parameterized by the size of Odd Cycle
Transversal (I.e. Bipartite + k vertices)
• Domset, FVS are hard even for constant k (as they are
NP-hard in bipartite graphs).
11
43. Parameterized by the size of Odd Cycle
Transversal (I.e. Bipartite + k vertices)
• Domset, FVS are hard even for constant k (as they are
NP-hard in bipartite graphs).
• For VC, we have seen it FPT. There is a randomized
polynomial sized kernel.
11
44. Parameterized by the size of Odd Cycle
Transversal (I.e. Bipartite + k vertices)
• Domset, FVS are hard even for constant k (as they are
NP-hard in bipartite graphs).
• For VC, we have seen it FPT. There is a randomized
polynomial sized kernel.
• 3-Coloring
11
45. Parameterized by the size of Odd Cycle
Transversal (I.e. Bipartite + k vertices)
• Domset, FVS are hard even for constant k (as they are
NP-hard in bipartite graphs).
• For VC, we have seen it FPT. There is a randomized
polynomial sized kernel.
• 3-Coloring is NP-hard in graphs that are Bipartite + 2
vertices and Bipartite + 3 edges (Cai 2003).
11
47. Parameterized by the length of the
longest odd cycle
Another way to measure “closeness” to bipartite graphs.
12
48. Parameterized by the length of the
longest odd cycle
Another way to measure “closeness” to bipartite graphs.
• Input: A graph G, k is the length of the longest cycle in G,
an integer
• Parameter: k
• Question: Does G have a VC/MIS/Chromatic number of
size at most or IS of size at least ?
12
49. Parameterized by the length of the
longest odd cycle
Another way to measure “closeness” to bipartite graphs.
• Input: A graph G, k is the length of the longest cycle in G,
an integer
• Parameter: k
• Question: Does G have a VC/MIS/Chromatic number of
size at most or IS of size at least ?
• MIS can be solved in time nO(k) – Hsu, Ikura and
Nemhauser, Math. Programming, 1981
• MAX-CUT can be solved in time nO(k) – Grötschel and
Nemhauser, Math. Programming, 1984
12
50. Parameterized by the length of the
longest odd cycle
Another way to measure “closeness” to bipartite graphs.
• Input: A graph G, k is the length of the longest cycle in G,
an integer
• Parameter: k
• Question: Does G have a VC/MIS/Chromatic number of
size at most or IS of size at least ?
• MIS can be solved in time nO(k) – Hsu, Ikura and
Nemhauser, Math. Programming, 1981
• MAX-CUT can be solved in time nO(k) – Grötschel and
Nemhauser, Math. Programming, 1984
• MIS, Max-Cut, Coloring can be solved in time 2O(k)nO(1).
(Rai and Panolan – 2012)
12
54. Other Highlights/Results/Open
Problems
• VC param by degree 3 modulator: Para-NP-Hard.
• FVS param by degree 3 modulator: Open
• CLIQUE param by VC: (i.e. Clique in Independent Set + k
vertices)
At most one vertex from V S can participate in the solution.
S
13
55. Other Highlights/Results/Open
Problems
• VC param by degree 3 modulator: Para-NP-Hard.
• FVS param by degree 3 modulator: Open
• CLIQUE param by VC: (i.e. Clique in Independent Set + k
vertices)
At most one vertex from V S can participate in the solution.
S
FPT but has no polynomial kernel unless NP ⊆ coNP/poly.
13
56. Parameter Ecology
(figure from Bart Jansen’s thesis)
Vertex Cover Max Leaf
Distance to
Linear Forest
Genus
Distance
to a Clique
Feedback
Vertex Set
Cutwidth Bandwidth
Topological
Bandwidth
Distance
to Chordal
Distance to
Outerplanar
Pathwidth
Odd Cycle
Transversal
Treewidth
Distance
to Perfect
Chromatic
Number
`2O(h2
)[72,84]
[39]
` h
1
` 2 O(h2
) [84]
h +
1
`[8]
` 2 O(
p h)
[118]
` h `2
[29, 30]
[8]h+
2
`
h + 2
`
` h + 1 [8]
Figure 1: A hierarchy of parameters, with larger parameters drawn higher. An unlabeled line
between two parameters means that the parameter drawn lowest is never larger than the one
drawn highest. These unlabeled relationships (cf. [109]) follow from inclusions between graph
classes [18] or bounds which can be found in Bodlaender’s survey on treewidth [8]. When a
line between parameters is labeled by a bound, the lower-drawn parameter is represented by `
and the higher-drawn parameter by h.
14
58. The general ecology program
• If a parameter is W-hard (or has no polykernel under
complexity theoretic assumptions) under some
parameterization, try a larger parameter.
15
59. The general ecology program
• If a parameter is W-hard (or has no polykernel under
complexity theoretic assumptions) under some
parameterization, try a larger parameter.
• If a parameter is FPT (or has a polykernel) under some
parameterization, try a smaller parameter.
15
60. The general ecology program
• If a parameter is W-hard (or has no polykernel under
complexity theoretic assumptions) under some
parameterization, try a larger parameter.
• If a parameter is FPT (or has a polykernel) under some
parameterization, try a smaller parameter.
15
61. The general ecology program
• If a parameter is W-hard (or has no polykernel under
complexity theoretic assumptions) under some
parameterization, try a larger parameter.
• If a parameter is FPT (or has a polykernel) under some
parameterization, try a smaller parameter.
Helps understand the role of parameters in the complexity of
the problem.
15
65. Input parameterizations within P
LINEAR PROGRAMMING (LP)
• LP in d-variables and n-constraints — nO(d) – Simplex
• 22O(d)
n – Megiddo 1984 (JACM)
16
66. Input parameterizations within P
LINEAR PROGRAMMING (LP)
• LP in d-variables and n-constraints — nO(d) – Simplex
• 22O(d)
n – Megiddo 1984 (JACM)
• d2n + 2O(
√
d log d)
– randomized – Combination of Clarkson
(JACM 1995)/ Kalai (STOC 1992) / Matoušek, Sharir, and
Welzl (Algorithmica 1996)
16
67. Input parameterizations within P
LINEAR PROGRAMMING (LP)
• LP in d-variables and n-constraints — nO(d) – Simplex
• 22O(d)
n – Megiddo 1984 (JACM)
• d2n + 2O(
√
d log d)
– randomized – Combination of Clarkson
(JACM 1995)/ Kalai (STOC 1992) / Matoušek, Sharir, and
Welzl (Algorithmica 1996)
•
√
d
O(d)
(log d)3dn – deterministic – Chan (SODA 16)
16
68. LP Developments
simplex method det. O(n/d)d/2+O(1)
Megiddo [24] det. 2O(2d
)
n
Clarkson [9]/Dyer [14] det. 3d2
n
Dyer and Frieze [15] rand. O(d)3d
(log d)d
n
Clarkson [10] rand. d2
n + O(d)d/2+O(1)
log n + d4
√
n log n
Seidel [26] rand. d!n
Kalai [19]/Matouˇsek, Sharir, and Welzl [23] rand. min{d2
2d
n, e2
√
d ln(n/
√
d)+O(
√
d+log n)
}
combination of [10] and [19, 23] rand. d2
n + 2O(
√
d log d)
Hansen and Zwick [18] rand. 2O(
√
d log((n−d)/d))
n
Agarwal, Sharir, and Toledo [4] det. O(d)10d
(log d)2d
n
Chazelle and Matouˇsek [8] det. O(d)7d
(log d)d
n
Br¨onnimann, Chazelle, and Matouˇsek [5] det. O(d)5d
(log d)d
n
this paper det. O(d)d/2
(log d)3d
n
Table 1: Deterministic and randomized time bounds for linear programming on the real RAM.
2. Clarkson’s paper [10] described a second sampling
algorithm, based on iterative reweighting (or in
more trendy terms, the multiplicative weights update
method). This second algorithm has expected n log n
running time in terms of n, but has better depen-
Given a set H of n hyperplanes in Rd
, find a
p that lies on or below all hyperplanes in H,
minimizing a given linear objective function.
All our algorithms rely on the concept of ε
Table taken from the SODA 16 paper of Chan
17
70. Recent work
• Longest path in interval graphs can be solved in O(n4) time,
but in O(k9n) time if the interval graph is a proper interval
graph + k vertices (GMN IPEC 2015).
18
71. Recent work
• Longest path in interval graphs can be solved in O(n4) time,
but in O(k9n) time if the interval graph is a proper interval
graph + k vertices (GMN IPEC 2015).
• Diameter in treewidth k graphs can be found in
2O(k lg k)n1+o(1) time, but can’t be solved in 2o(k)n2−ε time
under some plausible assumptions (AWW SODA 2016).
18
72. Recent work
• Longest path in interval graphs can be solved in O(n4) time,
but in O(k9n) time if the interval graph is a proper interval
graph + k vertices (GMN IPEC 2015).
• Diameter in treewidth k graphs can be found in
2O(k lg k)n1+o(1) time, but can’t be solved in 2o(k)n2−ε time
under some plausible assumptions (AWW SODA 2016).
• O(k3n lg2
n) randomized algorithm to find a maximum
matching in a graph of treewidth k (earlier there was a
O(2kn) algorithm) (FLPSW SODA 2017).
18
73. Some Surveys
• Towards fully multivariate algorithmics: Parameter ecology
and deconstruction of computational complexity, (Fellows,
Jansen and Rosamond, European Journal of Combinatorics
2013)
• Parameter Ecology for Feedback Vertex Set, (Jansen, R.
Vatshelle, Tsingua Science and Technology Journal on
information sciences, 2014).
• Parameterized SAT (Szeider, Encyclopedia of Algorithms
2016).
• Backdoors to Satisfaction (Gaspers and Szeider,
Multivariate Algorithmics Revolution and Beyond, 2012)
19
76. Decision versions of some NP-hard
problems
VC-perfect matching (k is the parameter)
• Input: A graph G on n vertices and m edges, that has a
perfect matching.
• Question: Does G have a vertex cover of size at most k?
21
77. Decision versions of some NP-hard
problems
VC-perfect matching (k is the parameter)
• Input: A graph G on n vertices and m edges, that has a
perfect matching.
• Question: Does G have a vertex cover of size at most k?
Max 3-SAT (k is the parameter)
• Input: A 3-CNF formula F on n variables and m clauses.
• Question: Does F have an assignment satisfying at least k
clauses?
21
78. Decision versions of some NP-hard
problems
VC-perfect matching (k is the parameter)
• Input: A graph G on n vertices and m edges, that has a
perfect matching.
• Question: Does G have a vertex cover of size at most k?
Max 3-SAT (k is the parameter)
• Input: A 3-CNF formula F on n variables and m clauses.
• Question: Does F have an assignment satisfying at least k
clauses?
MaxCut (k is the parameter)
• Input: A graph G on n vertices and m edges
• Question: Does G have a cut on at least k edges?
21
79. Decision versions of some NP-hard
problems
VC-perfect matching (k is the parameter)
• Input: A graph G on n vertices and m edges, that has a
perfect matching.
• Question: Does G have a vertex cover of size at most k?
Max 3-SAT (k is the parameter)
• Input: A 3-CNF formula F on n variables and m clauses.
• Question: Does F have an assignment satisfying at least k
clauses?
MaxCut (k is the parameter)
• Input: A graph G on n vertices and m edges
• Question: Does G have a cut on at least k edges?
Planar MIS (k is the parameter)
• Input: A planar graph G on n vertices and m edges.
• Question: Does G have an independent set of size at least
k?
21
82. Trivial linear kernels and hence trivial
FPT algorithms
Max-3SAT If k ≤ m/2, then YES, else m ≤ 2k (as every CNF sat on m
clauses has a satisfying assignment with at least m/2
clauses)
22
83. Trivial linear kernels and hence trivial
FPT algorithms
Max-3SAT If k ≤ m/2, then YES, else m ≤ 2k (as every CNF sat on m
clauses has a satisfying assignment with at least m/2
clauses)
MaxCUT If k ≤ m/2, then YES, else m ≤ 2k (as every graph has a cut
of size at least m/2)
22
84. Trivial linear kernels and hence trivial
FPT algorithms
Max-3SAT If k ≤ m/2, then YES, else m ≤ 2k (as every CNF sat on m
clauses has a satisfying assignment with at least m/2
clauses)
MaxCUT If k ≤ m/2, then YES, else m ≤ 2k (as every graph has a cut
of size at least m/2)
VC-PM If k ≤ n/2, then NO, else n ≤ 2k (as matching size is a
lower bound for vertex cover size)
22
85. Trivial linear kernels and hence trivial
FPT algorithms
Max-3SAT If k ≤ m/2, then YES, else m ≤ 2k (as every CNF sat on m
clauses has a satisfying assignment with at least m/2
clauses)
MaxCUT If k ≤ m/2, then YES, else m ≤ 2k (as every graph has a cut
of size at least m/2)
VC-PM If k ≤ n/2, then NO, else n ≤ 2k (as matching size is a
lower bound for vertex cover size)
lanar-MIS If k ≤ n/4, then YES, else n ≤ 4k (as a planar graph has an
independent set of size at least n/4)
22
87. What’s happening?
• The problems are trivial for small values of k (as there is a
large lower bound for their solution size),
23
88. What’s happening?
• The problems are trivial for small values of k (as there is a
large lower bound for their solution size), and when brute
force algorithm is applied, k is large.
23
89. What’s happening?
• The problems are trivial for small values of k (as there is a
large lower bound for their solution size), and when brute
force algorithm is applied, k is large.
• No new FPT algorithm
23
90. What’s happening?
• The problems are trivial for small values of k (as there is a
large lower bound for their solution size), and when brute
force algorithm is applied, k is large.
• No new FPT algorithm
• So a better way to parameterize would be
23
92. Above guarantee parameterization
AG Max 3-SAT (k is the parameter)
• Input: A 3-CNF formula F on n variables and m clauses.
• Question: Does F have an assignment satisfying at least
m/2 + k clauses?
24
93. Above guarantee parameterization
AG Max 3-SAT (k is the parameter)
• Input: A 3-CNF formula F on n variables and m clauses.
• Question: Does F have an assignment satisfying at least
m/2 + k clauses?
AG Max Cut (k is the parameter)
• Input: A graph G on n vertices and m edges
• Question: Does G have a cut on at least m/2 + k edges?
24
94. Above guarantee parameterization
AG Max 3-SAT (k is the parameter)
• Input: A 3-CNF formula F on n variables and m clauses.
• Question: Does F have an assignment satisfying at least
m/2 + k clauses?
AG Max Cut (k is the parameter)
• Input: A graph G on n vertices and m edges
• Question: Does G have a cut on at least m/2 + k edges?
AG VC-perfect matching (k is the parameter)
• Input: A graph G on n vertices with a perfect matching.
• Question: Does G have a vertex cover of size at most
n/2 + k?
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95. Above guarantee parameterization
AG Max 3-SAT (k is the parameter)
• Input: A 3-CNF formula F on n variables and m clauses.
• Question: Does F have an assignment satisfying at least
m/2 + k clauses?
AG Max Cut (k is the parameter)
• Input: A graph G on n vertices and m edges
• Question: Does G have a cut on at least m/2 + k edges?
AG VC-perfect matching (k is the parameter)
• Input: A graph G on n vertices with a perfect matching.
• Question: Does G have a vertex cover of size at most
n/2 + k?
Planar MIS (k is the parameter)
• Input: A planar graph G on n vertices and m edges.
• Question: Does G have an IS of size at least n/4 + k?
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96. Above guarantee parameterization -
FPT Results
AG-MAXSAT: It is FPT to determine whether there is an
assignment satisfying
• at least m/2 + k clauses in a CNF formula or
• at least k more than the expected solution of a random
assignment.
• at least k more than the number of variables
One can generalize some of these results for general CSPs.
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97. Above guarantee MaxCUT
AG-MAXCUT: It is FPT to determine whether there is a cut of
size
• at least m/2 + k (MR 1999);
• at least m/2 + (n − 1)/4 + k (CJM 2011); also has a
polynomial kernel, FPT bound is optimal under ETH.
• at least n − 1 + k (??)
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98. Above guarantee Vertex Cover
AGVC: It is FPT to determine whethere there is a VC of size
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99. Above guarantee Vertex Cover
AGVC: It is FPT to determine whethere there is a VC of size
• at most MM + k; (LNRRS ACM TALG 2014)
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100. Above guarantee Vertex Cover
AGVC: It is FPT to determine whethere there is a VC of size
• at most MM + k; (LNRRS ACM TALG 2014)
• at most LPopt + k (LNRRS ACM TALG 2014)
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101. Above guarantee Vertex Cover
AGVC: It is FPT to determine whethere there is a VC of size
• at most MM + k; (LNRRS ACM TALG 2014)
• at most LPopt + k (LNRRS ACM TALG 2014)
• at most 2LPopt − MM + k: O(3k · nO(1)) (GP SODA 2015).
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102. Planar MIS (k is the parameter)
• Input: A planar graph G on n vertices and m edges.
• Question: Does G have an IS of size at least n/4 + k?
FPT or W-hard?
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103. Planar MIS (k is the parameter)
• Input: A planar graph G on n vertices and m edges.
• Question: Does G have an IS of size at least n/4 + k?
FPT or W-hard? Famously OPEN for several years.
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104. Planar MIS (k is the parameter)
• Input: A planar graph G on n vertices and m edges.
• Question: Does G have an IS of size at least n/4 + k?
FPT or W-hard? Famously OPEN for several years.
• In triangle free planar graphs, one can find an independent
set of size n/3 + k in FPT time (ESA 2014).
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106. Above guarantee parameterization - W
hardness Results
• Does a given connected graph have a 2-connected subgraph
with at most n + k edges? (note than n is a guaranteed
lower bound).
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107. Above guarantee parameterization - W
hardness Results
• Does a given connected graph have a 2-connected subgraph
with at most n + k edges? (note than n is a guaranteed
lower bound).
• It is NP-hard even for k = 0
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108. Parameterizing from the extreme
• Input: A graph G
• Parameter: k
• Question: Does G have a vertex cover of size at most n − k?
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109. Parameterizing from the extreme
• Input: A graph G
• Parameter: k
• Question: Does G have a vertex cover of size at most n − k?
This is W[1]-hard as it is the same as k-IS.
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110. Parameterizing from the extreme
• Input: A graph G
• Parameter: k
• Question: Does G have a vertex cover of size at most n − k?
This is W[1]-hard as it is the same as k-IS.
• Input: A graph G
• Parameter: k
• Question: Does G have chromatic number at most n − k?
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111. Parameterizing from the extreme
• Input: A graph G
• Parameter: k
• Question: Does G have a vertex cover of size at most n − k?
This is W[1]-hard as it is the same as k-IS.
• Input: A graph G
• Parameter: k
• Question: Does G have chromatic number at most n − k?
This is FPT.
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112. Some Surveys
• Kernelization, constraint satisfaction problems
parameterized above average, Gregory Gutin, Encyclopedia
of Algorithms (2016).
• Kernelization, permutation CSPs parameterized above
average, Gregory Gutin, Encyclopedia of Algorithms
(2016).
• Constraint satisfaction problems parameterized above or
below tight bounds, a survey, Gregory Gutin and Anders
Yeo, in the Multivariate algorithmic revolution and beyond,
2012.
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113. Above guarantee parameterization
within P
Dominating set in tournaments
• Input: A tournament T on n vertices.
• Parameter: k
• Question: Does G have a dominating on k vertices?
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114. Above guarantee parameterization
within P
Dominating set in tournaments
• Input: A tournament T on n vertices.
• Parameter: k
• Question: Does G have a dominating on k vertices?
This is W[2]-hard (as in the case of general directed graphs).
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115. Above guarantee parameterization
within P
Dominating set in tournaments
• Input: A tournament T on n vertices.
• Parameter: k
• Question: Does G have a dominating on k vertices?
This is W[2]-hard (as in the case of general directed graphs).
However, every tournament on n vertices has a dominating set
of size at least lg n and can be found in O(n2) time.
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116. Above guarantee parameterization
within P
Dominating set in tournaments
• Input: A tournament T on n vertices.
• Parameter: k
• Question: Does G have a dominating on k vertices?
This is W[2]-hard (as in the case of general directed graphs).
However, every tournament on n vertices has a dominating set
of size at least lg n and can be found in O(n2) time.
• Input: A tournament T on n vertices.
• Parameter: k
• Question: Does G have a dominating on lg n − k vertices?
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117. Above guarantee parameterization
within P
Dominating set in tournaments
• Input: A tournament T on n vertices.
• Parameter: k
• Question: Does G have a dominating on k vertices?
This is W[2]-hard (as in the case of general directed graphs).
However, every tournament on n vertices has a dominating set
of size at least lg n and can be found in O(n2) time.
• Input: A tournament T on n vertices.
• Parameter: k
• Question: Does G have a dominating on lg n − k vertices?
OPEN
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118. Finding lg n + k sized dominating sets
• Input: A tournament T on n vertices.
• Parameter: k
• Question: Find a dominating set on lg n + k vertices?
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119. Finding lg n + k sized dominating sets
• Input: A tournament T on n vertices.
• Parameter: k
• Question: Find a dominating set on lg n + k vertices?
Can be done in O(n2/k) time, and the bound is optimal.
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120. Finding lg n + k sized dominating sets
• Input: A tournament T on n vertices.
• Parameter: k
• Question: Find a dominating set on lg n + k vertices?
Can be done in O(n2/k) time, and the bound is optimal.
Can be generalized similarly for distance d-dominating sets.
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121. Finding lg n + k sized dominating sets
• Input: A tournament T on n vertices.
• Parameter: k
• Question: Find a dominating set on lg n + k vertices?
Can be done in O(n2/k) time, and the bound is optimal.
Can be generalized similarly for distance d-dominating sets.
Theorem: Any dominating set has a distance 2-dominating set
of size 1 (called a king) that can be found in O(n3/2) time.
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122. Finding lg n + k sized dominating sets
• Input: A tournament T on n vertices.
• Parameter: k
• Question: Find a dominating set on lg n + k vertices?
Can be done in O(n2/k) time, and the bound is optimal.
Can be generalized similarly for distance d-dominating sets.
Theorem: Any dominating set has a distance 2-dominating set
of size 1 (called a king) that can be found in O(n3/2) time. We
can find a 2-dominating set of size lg n + k in O((n3/2)/k) time
and the bound is optimal.
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124. To Conclude
• We have come a long way in making researchers consider
parameterized complexity as a way to deal with
NP-completeness.
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125. To Conclude
• We have come a long way in making researchers consider
parameterized complexity as a way to deal with
NP-completeness.
• Hopefully this talk will inspire researchers to investigate
problems (not necessarily NP-complete) through multiple
different parameterizations!
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126. To Conclude
• We have come a long way in making researchers consider
parameterized complexity as a way to deal with
NP-completeness.
• Hopefully this talk will inspire researchers to investigate
problems (not necessarily NP-complete) through multiple
different parameterizations!
• Some general dichotomy theorems?
34