Covering Small Independent Sets and Separators (with ...aritra/...Bhubaneshwar10022019.pdf · Sets...

Covering Small Independent Sets and Separators(with Applications)

Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Roohani Sharma, Meirav Zehavi

Recent Advances in AlgorithmsNISER BhubaneswarFebruary 10th, 2019

Table of Contents

1. Introduction/Literature2. Tool 13. Tool 24. Applications5. Concluding Remarks

Table of Contents

1. A Combinatorial Question - Tool 12. Applications of Tool 13. Stumble upon Literature4. Resolve some open questions using Tool 15. Need for the design of Tool 26. Design of Tool 27. Concluding Remarks

Behind the

scenes

n edges

F (G,1) 2 F (G,2) n + 2

F (G,3)

F (G,k)

Can the dependence of k be removed from the exponent on n? ?

2

✓n

2

◆+ 2

2k✓n

k

◆+ 2

⇡ nO(k)

Given a graph G and an integer k, an independent set covering family (ISCF) for (G,k) is a family of independent sets of G, say F (G,k), such that for any independent set X of G of size at most k, there exists Y ⊆ F (G,k), such that X ⊆ Y.

Independent Set Covering Lemma (ISCL)If G is d-degenerate, then for any k, there is an ISCF for (G,k) of size

2O(k log kd) log n.

In fact, such a family can be found in 2O(k log kd) (n+m). log n time.

Tool 1:

Towards Randomized Independent Set Covering Lemma

Given: A d-degenerate graph G, an integer kOutput: An independent set Y such that for any independent set X of size at most k, the Pr( ) �

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X ✓ Y<latexit sha1_base64="D/2I2LpEzfh36C0kV3Lt63nfKNQ=">AAAB9HicdVDJSgNBEO2JW4xb1KOXxiB4Cj2J2W5BLx4jGBNJhtDTqUma9Czp7gmEId/hxYMiXv0Yb/6NnUVQ0QcFj/eqqKrnRoIrTciHlVpb39jcSm9ndnb39g+yh0d3KowlgyYLRSjbLlUgeABNzbWAdiSB+q6Alju6mvutCUjFw+BWTyNwfDoIuMcZ1UZy2rirYleBhjG+72VzJE/KpVqRYJIvEbtSqxlCSLlaLGDbkDlyaIVGL/ve7Ycs9iHQTFClOjaJtJNQqTkTMMt0YwURZSM6gI6hAfVBOcni6Bk+M0ofe6E0FWi8UL9PJNRXauq7ptOneqh+e3PxL68Ta6/qJDyIYg0BWy7yYoF1iOcJ4D6XwLSYGkKZ5OZWzIZUUqZNThkTwten+H9yV8jbxTy5ucjVL1dxpNEJOkXnyEYVVEfXqIGaiKExekBP6NmaWI/Wi/W6bE1Zq5lj9APW2yea75IA</latexit>

1

2k(d+1)<latexit sha1_base64="W5aNk24gFMkkx6MtpUXh3us0i7w=">AAAB/XicdVDJSgNBEO2JW4xbXG5eGoMQEUJPYrZb0IvHCGaBZAw9PT1Jk56F7h4hDoO/4sWDIl79D2/+jZ1FUNEHBY/3qqiqZ4ecSYXQh5FaWl5ZXUuvZzY2t7Z3srt7bRlEgtAWCXggujaWlDOfthRTnHZDQbFnc9qxxxdTv3NLhWSBf60mIbU8PPSZywhWWhpkD/quwCQ2k7h4E4/zzql5kiSDbA4VUKVcLyGICmVkVut1TRCq1EpFaGoyRQ4s0Bxk3/tOQCKP+opwLGXPRKGyYiwUI5wmmX4kaYjJGA9pT1Mfe1Ra8ez6BB5rxYFuIHT5Cs7U7xMx9qSceLbu9LAayd/eVPzL60XKrVkx88NIUZ/MF7kRhyqA0yigwwQlik80wUQwfSskI6zjUDqwjA7h61P4P2kXC2apgK7Oco3zRRxpcAiOQB6YoAoa4BI0QQsQcAcewBN4Nu6NR+PFeJ23pozFzD74AePtE+DHlNs=</latexit>

Goal

For each vertex v ∈ V(G), colour it either red or blue, uniformly at random.Experiment

Graph G





1


Goal


Graph G

Graph G

RED = set of all vertices that are coloured redBLUE = set of all vertices that are coloured blue

GOOD EVENT = RED contains all vertices of X and none of its forward neighbours (i.e. all the forward neighbours of X are in BLUE)

Claim : If GOOD EVENT happens, then X ⊆ IND_RED

IND_RED= {v: v ∈ RED and all its forward neighbours in BLUE}

Graph G

Pr(GOOD EVENT) � 1

2|X|1

2|Nf (X)| �1

2k(d+1)





1


Goal






1


Goal


color v red with probability

color v blue with probability

1

d+ 1

d

d+ 1

1

2O(k log kd)

Randomized Independent Set Covering Lemma

Randomized Independent Set Covering Lemma (ISCL)

There is an algorithm that given a d-degenerate graph G and an integer k, outputs a family F (G,k) such that:• F (G,k) is an ISCF for (G,k) with probability at least 1- 1/n,

• |F (G,k)| ≤ 2O(k log kd) log n• Running time of the algorithm is O(|F (G,k)| . (n+m)).




1

2O(k log kd)

Deterministic Independent Set Covering Lemma

|U| = nFamily of functions {f1,…,ft}

fi : U -> [q]For each S ⊆ U, |S| ≤ l,

there exists some fi such that fi is injective on S

U

fi

1

2

q

(n,l,q)-perfect hash family, (q ≥ l)

Deterministic Independent Set Covering Lemma (n,l,q)-perfect hash family, (q ≥ l)


fi : U -> [q]For each S ⊆ U, |S| ≤ l,


U

fi

1

2

q

Deterministic Independent Set Covering Lemma (n,l,q)-perfect hash family


fi : U -> [q]

For each S ⊆ U, |S| ≤ l,


U

fi

1

2

q

Fredman, Komlos, Szemeredi [J. ACM ‘84]

For any n,l, a (n,l, lO(1))-perfect hash family of size lO(1) log n can be computed in time

lO(1) n log n.

Compute for l = k+kd

Deterministic Independent Set Covering Lemma (ISCL)

There is an algorithm that given a d-degenerate graph G and an integer k, runs in time 2O(k log kd) (n+m) log n, and outputs an ISCF for (G,k) of size 2O(k log kd) log n.

Applications: Design of Fixed-Parameter

Tractable Algorithms

Vertex Deletion Problems

Input: A graph G , an integer kQuestion: Does there exist a set of at most k vertices, say S, such that G-S has a property 𝝥?

• Feedback Vertex Set (FVS): 𝝥 is a forest.

• Odd Cycle Transversal (OCT): 𝝥 is a bipartite graph.

• Planar Vertex Deletion (PVD): 𝝥 is a planar graph.

• s-t Separator: 𝝥 is no path from s to t.• …

Conflict-free Vertex Deletion Problems

Input: A graph G , an integer kQuestion: Does there exist a set of at most k vertices, say S, such that G-S has a property 𝝥 and S is conflict-free (independent set)?

• Conflict-free Feedback Vertex Set (FVS)• Conflict-free Odd Cycle Transversal (OCT)• Conflict-free Planar Vertex Deletion (PVD)• Conflict-free s-t Separator• …

“Reusing” algorithms of vertex deletion problems to design algorithms for

Conflict-free Vertex Deletion Problems

FPT Algorithms Conflict-free Vertex Deletion Problems on d-degenerate

graphs (using ISCL)

Deterministic Independent Set Covering Lemma (ISCL)There is an algorithm that given a d-degenerate graph G and an integer k,

runs in time 2O(k log d) (n+m) log n, and outputs an ISCF for (G,k) of size

2O(k log d) log n.

Conflict-free s-t Separator on d-degenerate graphs

Conflict-free s-t Separator on d-degenerate graphs

Input: A graph G, an integer k, vertices s and tQuestion: Does there exist a set S, such that |S| ≤ k, S is an independent set in G and G-S has no path from s to t.

• Compute ISCF for (G,k), say F = {Y1, … , Yt}, where t = 2O(k log kd) log n (from

ISCL). Time Taken: 2O(k log kd) (n+m) log n

Input: A graph G, an integer k, vertices s and t, Y ⊆ V(G)

Question: Does there exist a set S, such that |S| ≤ k, S ⊆ Y and G-S has no path

from s to t.

Annotated s-t Separator

(G,k,s,t)

(G,k,s,t,Y1) (G,k,s,t,Y2) (G,k,s,t,Yt)

is a YES instance

if and only if

one of them is a YES instance

Annotated s-t Separator Input: A graph G, an integer k, vertices s and t, Y ⊆ V(G)

Question: Does there exist a set S, such that |S| ≤ k, S ⊆ Y and G-S has no path

from s to t.

Input: A graph G, an integer k, vertices s and t, w : V(G) -> N Question: Does there exist a set S, such that |S| ≤ k, w(S) ≤ k and G-S has no path from s to t.

Weighted s-t Separator

Assign weights to vertices, w(v) =1 if v ∈ Y, otherwise w(v) = k+1.

Annotated s-t Separator can be solved in O(k . (n+m)) time.

Conflict-free s-t Separator on d-degenerate graphs can be solved in 2O(k log kd) (n+m) log n time.

FPT Algorithms Conflict-free Vertex Deletion Problems on general

graphs

Conflict-free Feedback Vertex Set on general graphs

Approximate Feedback Vertex set, |X| ≤ c k

Forest,R = V(G) \ X

Compute ISCF for (G[X],k), say F2

|F2| = 2O(k)

(Using Brute force)

Compute ISCF for (G[R],k), say F1

|F1| = 2O(k log k) log n

(using ISCL)

F = {Y∪Z : Y∈F1, Z∈F2} is an ISCF for G.

|F | = 2kO(1)

log n

Open Problem at Dagstuhl Seminar Structure Theory and FPT Algorithms for Graphs, Digraphs and Hypergraphs

2007

of P-sequences, determine whether there is a common supersequence of length at most n+ k.This problem is at least as hard as directed feedback vertex set [FHS03], which was recentlysolved (and presented at this workshop) [CLL07, RS07].

References

[CLL07] Jianer Chen, Yang Liu, and Songjian Lu. Directed feedback vertex set problemis FPT. Preprint, 2007. http://kathrin.dagstuhl.de/files/Materials/07/07281/07281.ChenJianer.Paper.pdf

[FHS03] Michael Fellows, Michael Hallett, and Ulrike Stege. Analogs & duals of theMAST problem for sequences & trees. Journal of Algorithms 49:192–216, 2003.doi:10.1016/S0196-6774(03)00081-6

[RS07] Igor Razgon and Barry O’Sullivan. Directed feedback vertex set is fixed-parametertractable. Preprint, arXiv:0707.0282 [cs.DS], 2007.

Almost 2-coloringHenning FernauU. [email protected]

Is the following problem fixed-parameter tractable? Given a graph G and a parameter k,determine whether G has a vertex 3-coloring such that one color class has at most k vertices.In other words, the goal is to remove an independent set of k vertices such that the remaininggraph is bipartite.

Even Set / Minimum Distance in Linear CodesMichael FellowsU. [email protected]

Is the following problem fixed-parameter tractable? Given a graph G and a parameter k,determine whether there is a set S of between 1 and k vertices such that, for every vertex vin G, |N [v] \ S| is even. (Here N [v] denotes the set of neighbors of v in G.)

Almost 2-SATMichael FellowsU. [email protected]

Delete k Clauses 2-SAT is the following problem: given a 2-SAT formula, can you delete kclauses to make it satisfiable? This problem is equivalent in the fixed-parameter-tractabilitysense to the following problem: given a graph G having a perfect matching (so its minimumvertex cover has size at least n/2), does G have a vertex cover of size at most n/2 + k? Arethese problems fixed-parameter tractable?

3

Open Problem at Dagstuhl Seminar Structure Theory and FPT Algorithms for Graphs, Digraphs and Hypergraphs

2007

of P-sequences, determine whether there is a common supersequence of length at most n+ k.This problem is at least as hard as directed feedback vertex set [FHS03], which was recentlysolved (and presented at this workshop) [CLL07, RS07].

References

[CLL07] Jianer Chen, Yang Liu, and Songjian Lu. Directed feedback vertex set problemis FPT. Preprint, 2007. http://kathrin.dagstuhl.de/files/Materials/07/07281/07281.ChenJianer.Paper.pdf

[FHS03] Michael Fellows, Michael Hallett, and Ulrike Stege. Analogs & duals of theMAST problem for sequences & trees. Journal of Algorithms 49:192–216, 2003.doi:10.1016/S0196-6774(03)00081-6

[RS07] Igor Razgon and Barry O’Sullivan. Directed feedback vertex set is fixed-parametertractable. Preprint, arXiv:0707.0282 [cs.DS], 2007.

Almost 2-coloringHenning FernauU. [email protected]

Is the following problem fixed-parameter tractable? Given a graph G and a parameter k,determine whether G has a vertex 3-coloring such that one color class has at most k vertices.In other words, the goal is to remove an independent set of k vertices such that the remaininggraph is bipartite.

Even Set / Minimum Distance in Linear CodesMichael FellowsU. [email protected]

Is the following problem fixed-parameter tractable? Given a graph G and a parameter k,determine whether there is a set S of between 1 and k vertices such that, for every vertex vin G, |N [v] \ S| is even. (Here N [v] denotes the set of neighbors of v in G.)

Almost 2-SATMichael FellowsU. [email protected]

Delete k Clauses 2-SAT is the following problem: given a 2-SAT formula, can you delete kclauses to make it satisfiable? This problem is equivalent in the fixed-parameter-tractabilitysense to the following problem: given a graph G having a perfect matching (so its minimumvertex cover has size at least n/2), does G have a vertex cover of size at most n/2 + k? Arethese problems fixed-parameter tractable?

3

Is Conflict-free Odd Cycle Transversal FPT?Reed, Smith, Vetta : Finding odd cycle transversals.

[Operations Research Letters] (2004)❖

Is Conflict-free s-t Separator FPT?

Is Conflict-free s-t Separator FPT?Marx, O’Sullivan, Razgon : Finding small separators in linear time via

treewidth reduction. [ACM Trans. Algorithms] (2013)❖

Yes! 22kO(1)

(n+m)

Open Problems from Marx et al.1.Is it possible to improve the dependence of k to 2k

O(1) ?

2.Is Conflict-free Multicut FPT?

1.Yes! 2kO(1)

(n+m) log n

2.Yes! 2O(k3) n3 (n+m)

Lokshtanov, Panolan, Saurabh, S., Zehavi: Covering small independent sets and separators with applications to parameterized algorithms [SODA] (2018)

❖

Overview of Marx et el [TALG 2013] approach

Is Conflict-free s-t Separator FPT?

1. Treewidth Reduction Step2. Dynamic Programming on bounded treewidth graph

(G,k,s,t)

Treewidth Reduction Step

(G,k,s,t) (G’,k,s,t)f(k). (n+m) time

preserves all minimal s-t separators of size

at most k

treewidth(G’) = 2kO(1)

G’ is an “induced subgraph” of G

Dynamic Programming on bounded treewidth graph (G’)

Time taken: exponential in treewidth ⇒ 22kO(1)

(n+m)

Our approach [SODA 2018]1. Treewidth Reduction Step2. Dynamic Programming on bounded treewidth graph

Treewidth Reduction Step

(G,k,s,t) (G’,k,s,t)f(k). (n+m) time

preserves all minimal s-t separators of size

at most k

treewidth(G’) = 2kO(1)

G’ is an “induced subgraph” of G

Dynamic Programming on bounded treewidth graph (G’)

ISCL

ISCL on (G’,k)degeneracy(G’) ≤ treewidth(G’) = 2kO(1)

Conflict-free s-t Separator on d-degenerate graphs can be solved in 2O(k log kd) (n+m) log n time.

Conflict-free s-t Separator on general graphs can be solved in

2kO(1)

(n+m) log n time.

Conflict-free MulticutInput: A graph G, an integer k, terminal pairs T={(s1,t1), … , (sp,tp)}

Question: Does there exist a set S, such that |S| ≤ k, S is an independent set in G and, for all i∈{1,…,p}, there is no path from si to ti in G-s.

s1

s2

s3

t1

t2

T3

Tool 2: Degeneracy Reduction Preserving Minimal Multicuts

There exists a polynomial time algorithm that given a graph G, a set of terminal pairs T={(s1,t1), … , (sp,tp)} and an integer k, returns an

induced subgraph G’ of G and T’ ⊆ T such that:• Every minimal multicut of T in G of size at most k is a minimal multicut of T’ in G’,

• Every minimal multicut of T’ in G’ of size at most k is a minimal multicut of T in G.

• Degeneracy of G’ is 2O(k).

(G,T,k) (G’,T’,k)

Concluding Remarks

Extensions:ISCL for nowhere dense graphs

Barriers:• ISCL for general graphs• Induced Matching Covering on 1-degenerate graphs• Acyclic Subgraphs Covering on 2-degenerate graphs• r-scattered sets covering on 1-degenerate graphs

• ISCL beyond nowhere dense graphs?• Covering other families

Open:?

Thank you!

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