kernelization algorithms for graph and other structure modiﬁcation problems

Kernelization algorithms for graph and otherstructure modification problems

Anthony PEREZ

Supervisors: Stephane BESSY and Christophe PAUL

(AlGCo Team)

November 14

INTRODUCTION

(Graph) Modification problems

Input: A graph (or another structure) and a (graph) property G.Output: A minimum number of modification of the graph in order tosatisfy G.

modification: adding edges, deleting edges, deleting vertices, ...

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INTRODUCTION


CLUSTER EDITING

Input: A graph G = (V ,E).Output: A set F ⊆ (V × V ) of minimum size such that the graphH = (V ,E M F ) is a disjoint union of cliques.

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INTRODUCTION


Cover a broad range of NP-Hard problems:

VERTEX COVER

FEEDBACK VERTEX SET

More general: F -MINOR DELETION

EDGE-MULTICUT

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INTRODUCTION


Find applications in various domains:

bioinformaticsmachine learningrelational databasesimage processing

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INTRODUCTION

Different approaches

Most modification problems are NP-hard.How to solve them efficiently?

Approximation algorithmsExact exponential algorithmsPreprocessing steps (heuristics)

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INTRODUCTION




How to measure the efficiency of heuristics?

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INTRODUCTION




Exploit the fact that the number of modifications needed should besmall compared to the instance size n.

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Outline of the talk

1 Parameterized complexity

Part I. Graph Modification Problems

2 Branches and generic reduction rules

3 PROPER INTERVAL COMPLETION

Part II. Different modification problems

4 Considered problems

5 FEEDBACK ARC SET IN TOURNAMENTS

PARAMETERIZED COMPLEXITY

Parameterized problem

G-MODIFICATION

Input: A graph G = (V ,E), k ∈ N.Parameter: k .Output: A set F ⊆ (V × V ) of size at most k such that the graphH = (V ,E M F ) belongs to G.

Idea. Measure the complexity of a problem with respect tosome parameter k .

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Parameterized problem

G-MODIFICATION

Input: A graph G = (V ,E), k ∈ N.Parameter: k .Output: A set F ⊆ (V × V ) of size at most k such that the graphH = (V ,E M F ) belongs to G.

Parameterized algorithmA problem parameterized by some k ∈ N admits a parameterizedalgorithm if it can be solved in time f (k) · nO(1).

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Kernels

Given an instance (I, k) of a parameterized problem L,a kernelization algorithm:

runs in time Poly(|I|+ k)

and outputs an instance (I′, k ′) such that:(i) (I, k) ∈ YES ⇔ (I′, k ′) ∈ YES(ii) |I′| 6 h(k) and k ′ 6 k

(I, k) (I ′, k ′)

|I ′| 6 h(k)k ′ 6 k

Poly(|I|+ k)

Do all parameterized problems admit polynomial kernels?

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Kernels




Theorem (Folklore)Parameterized algorithm⇔ Kernelization algorithm


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Kernels




Size: super-polynomial


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Lower bounds for kernels

There exist some parameterized problems that do not admit polynomialkernels. (under a complexity-theoretic assumption)

(i) Or-composition [Bodlaender et al., 2008 - Fortnow and Santhanam, 2008]

(ii) Polynomial time and parameter transformations[Bodlaender et al., 2009]

(iii) Cross-composition [Bodlaender et al., 2011]

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Graph modification problems

2 Branches and generic reduction rules


G-MODIFICATION

Input: A graph G = (V ,E), k ∈ N.

Parameter: k .

Output: A set F ⊆ (V × V ) of size at most k s.t. the graph H = (V ,E M F ) belongs to G.

BRANCHES AND GENERIC REDUCTION RULES PROPER INTERVAL COMPLETION

Generic reduction rules

Connected component.

If G is hereditary and closed under disjoint union, remove anyconnected component C that belongs to G.

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Sunflower.

Consider a finite forbidden induced subgraph of G (obstruction).For any pair e ⊆ (V ×V ) that belongs to a set of k + 1 obstructionspairwise intersecting exactly in e, transform G into (V ,E M {e}).

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Generic reduction rulesCritical clique.

Assume G is hereditary and closed under true twin addition.For any critical clique T with |T | > k + 1, remove |T | − (k + 1)arbitrary vertices from T .

v

u

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Generic reduction rulesCritical clique.

Assume G is hereditary and closed under true twin addition.For any critical clique T with |T | > k + 1, remove |T | − (k + 1)arbitrary vertices from T .

Lemma [Bessy, Paul and P., 2010]

There always exists an optimal editionthat preserves the critical cliques.

k = 1

k = 1

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Branches: a natural idea

Reduce set of vertices that induce a graph belonging to G.The Connected Component rule is a Branch reduction rule.

Context: can be used on problems where G admits a so-calledadjacency decomposition.

Branch: set of vertices B ⊆ Vsuch that:

(i) G[B] ∈ G and,(ii) B is connected properly

to the rest of the graph.

B

G[B] ∈ G

G \ B

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Outline

2 Branches and generic reduction rulesGeneric reduction rulesBranches


Definition and known resultsBranchesReducing the branches


DefinitionPROPER INTERVAL COMPLETION

Input: A graph G = (V ,E), k ∈ N.Parameter: k .Output: A set F ⊆ (V × V ) \ E of size at most k such thatH = (V ,E ∪ F ) is a proper interval graph.

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NP-Complete [Golumbic et al., 1994]

FPT : O(24km) (motivated by applications in genomic research)[Kaplan, Shamir and Tarjan, 1994]

Polynomial kernel?

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Theorem [Bessy and P., 2011]

The PROPER INTERVAL COMPLETION problem admits a kernel withO(k4) vertices.

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Some useful results

A graph is a proper interval graph if and only if:

it does not contain any of the following graphs as an inducedsubgraph.

claw p-cycle (p ≥ 4)3-sun net

[Wegner, 1967]

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Some useful results

A graph is a proper interval graph if and only if:

its vertices admit an ordering v1 . . . vn such that:

vivj ∈ E i < j ⇒ vivl , vlvj ∈ E , i < l < j

[Looges and Olartu, 1993]

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Remarks. Proper interval graphs are hereditary and:

(i) closed under disjoint union:the Connected Component rule can be applied.

(ii) do not admit any claw or C4 as an induced subgraph:the Sunflower rule can be applied.

(iii) closed under true twin addition:the Critical Clique rule can be applied.

What about branches?

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Adjacency decomposition

2 3 4 5 6 7 8 9(b)

(a)

9

2

1

4

3

5

6

7

8

7895

4321

6

1

Branches can be used on PROPER INTERVAL COMPLETION.

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How to define a branch?

Consider the structure of a solution.Look at unaffected vertices.

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Branches

B1 BR B2

L b1 R C

B

bl

A subset B of V is a branch if:

(i) G[B] is a connected PIG with umbrella ordering σB = b1, . . . ,b|B|,(ii) The vertex set V \ B can be partitioned into sets L,R and C with:

no edges between B and Cevery vertex in L (resp. R) has a neighbor in BNB(L) ⊂ NB[b1] = {b1, . . . ,bl′}NB(R) ⊂ NB[b|B|] = {bl , . . . ,b|B|}NL(bi+1) ⊆ NL(bi) for every 1 ≤ i < l ′ and NR(bi) ⊆ NR(bi+1) forevery l ≤ i < |B|

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Branches

B1 BR B2

L b1 b|B| R C

B




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Branches

B1 BR B2

L b1 b|B| R C

B

blbl ′




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Branches

B1 BR B2

L b1 b|B| R C

B

bl ′ bl




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Branches

B1 BR B2

L b|B| R C

B

b1 bl ′ bl




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Branches

B1 BR B2

L b1 R C

B

bl

If L = ∅ (or R = ∅), B is a 1-branch, otherwise B is a 2-branchIf B is a clique, we call B a K-join

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Outline

2 Branches and generic reduction rulesGeneric reduction rulesBranches


Definition and known resultsBranchesReducing the branches


Reducing the K -joins

Cannot be done directly.

x y z t

Assuming the graph is reduced by the generic rules, we can removeO(k3) vertices from any K -join to obtain a clean K -join.

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Reducing the K -joins

Cannot be done directly.

A clean K -join does not intersect any claw or C4.

Assuming the graph is reduced by the generic rules, we can removeO(k3) vertices from any K -join to obtain a clean K -join.

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Reducing the clean K -joins

Let B be a clean K -join of size at least 2k + 2. Let Bf be the k + 1 firstvertices of B, Bl be its k + 1 last vertices and M = B \ (Bf ∪ Bl).Remove the set of vertices M from G.

Bl (k + 1 vertices)MBf (k + 1 vertices)

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Reducing the clean K -joins

Let B be a clean K -join of size at least 2k + 2. Let Bf be the k + 1 firstvertices of B, Bl be its k + 1 last vertices and M = B \ (Bf ∪ Bl).Remove the set of vertices M from G.

Can be carried out in polynomial time!

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Reducing the branches

In polynomial time, the 1- and 2-branches can be reduced to O(k3)vertices.

B1

Remove

B

BR

2k + 1 vertices

R G \ (B ∪ R)

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Reducing the branches

In polynomial time, the 1- and 2-branches can be reduced to O(k3)vertices.

B1

Remove

B

BR

2k + 1 vertices

R G \ (B ∪ R)

Remove

B

2k + 1 vertices

BR

2k + 1 vertices

B1

LB2

RB′2B′1

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Main result



1-branch 1-branch2-branchK -join K -join K -join

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Main result



O(k3) O(k3) O(k3) O(k3) O(k3)O(k3)

1-branch 1-branch2-branchK -join K -join K -join

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Main result



Related result [Bessy, Paul and P., 2010]

The CLOSEST 3-LEAF POWER problem admits a kernel with O(k3)vertices.

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Different modification problems

4 Considered problems


Π-EDITION

Input: A dense set R of p-sized relations defined over an universe V , an integer k ∈ N.

Parameter: k .

Output: A set F ⊆ R of size at most k whose modification satisfies Π.

CONSIDERED PROBLEMS FEEDBACK ARC SET IN TOURNAMENTS

FEEDBACK ARC SET IN TOURNAMENTS (FAST)

Input: A tournament T = (V ,A) and an integer k ∈ N.Parameter: k .Output: A set at most k arcs whose reversal results in an acyclictournament.

23 4 21

3

1 4

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FEEDBACK ARC SET IN TOURNAMENTS (FAST)

Input: A tournament T = (V ,A) and an integer k ∈ N.Parameter: k .Output: A set at most k arcs whose reversal results in an acyclictournament.

NP-Complete [Charbit et al., 2007]

Admits constant-factor approximation algorithms [Kenyon-Mathieu andSchudy, 2007]

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DENSE ROOTED TRIPLET INCONSISTENCY (RTI)

Input: A set of leaves V and a dense collection R of rooted binarytrees over three leaves of V .Parameter: k .Output: A set of at most k triplets whose modification leads to acollection admitting a consistent rooted binary tree defined over V .

ab bd dc cca b da

t1 t2 t3 t4

R := {ab|c, cd |b, ab|d , ac|d}R := {t1, t2, t3, t4}

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a ab bd d dc cca b

a b c d

t1 t2 t3 t4

R := {ab|c, cd |b, ab|d , ac|d}

T is not consistent withR

R := {t1, t2, t3, t4}

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a cb bd d ac dca b

a b c d

t1 t2 t3 t4

R := {ab|c, cd |b, ab|d , cd |a}

T is consistent withR

R := {t1, t2, t3, t4}

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NP-Complete [Barky et al., 2010]

Does not admit a constant-factor approximation algorithm yet

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Outline

4 Considered problemsFEEDBACK ARC SET IN TOURNAMENTS

DENSE ROOTED TRIPLET INCONSISTENCY

Conflict Packing


Reduction rulesConflict Packing


Consistency

FAST (folklore)The following properties are equivalent:

(i) T is acyclic(ii) T does not contain any directed triangle

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Consistency

RTI [Guillemot and Mnich, 2010]

The following properties are equivalent:(i) R is consistent(ii) R does not contain any conflict on four leaves

Conflict. Set of vertices C ⊆ V that does not admit a consistent rooted binary tree.

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Parameterized complexity

FAST RTIFPT O∗(2

√k log k ) a FPT O∗(2k1/3 log k ) b

Kernel with O(k2) verticesa Kernel with O(k2) verticesb

Linear vertex-kernel c No such result known before.

a[Alon et al., 2009]b[Guillemot and Mnich, 2010]c[Bessy et al., 2009]

The linear vertex-kernel for FAST described by [Bessy et al., 2009]uses a constant-factor approximation algorithm.Their reduction rules can be adapted to RTI.But no constant-factor approximation!

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Conflict Packing

[Paul, P. and Thomasse, 2011]

works on problems characterized by some finite conflicts.maximal collection of p-uplets disjoint conflits C.provides a lower bound on the number of modification required.implies that the instance induced by V \ V (C) is consistent.

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Reduction rules

Remove any vertex that is not part of any directed triangle. a.acan be carried out in polynomial time.

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Reduction rules

Safe partition

Assume V (T ) is ordered under some ordering σ, and let P be apartition of σ into intervals.

Vl

AI := {uv ∈ A | ∃ i u, v ∈ Vi}

V1 V2

AO := A \ AI

B is the set of backward arcs of AO (arcs vivj with i > j).

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Reduction rules

Safe partition

P is safe if there exist |B| arc-disjoint conflicts using arcs of AOonly.

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Safe Partition Reduction Rule

[Bessy et al., 2009]

Let P be a safe partition of an ordered tournament T = (V ,A, σ).Reverse every arc of B and decrease k accordingly.

Use constant-factor approximation algorithm.Use Conflict Packing.

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Safe Partition Reduction Rule

[Bessy et al., 2009]

Let P be a safe partition of an ordered tournament T = (V ,A, σ).Reverse every arc of B and decrease k accordingly.

Main questionHow to compute a safe partition in polynomial time?

Use constant-factor approximation algorithm.Use Conflict Packing.

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Conflict Packing

A conflict packing of a tournament is a maximal collection ofarc-disjoint directed triangles.

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Conflict Packing


Can be computed greedily (i.e. in polynomial time).Let C be a conflict packing. If T = (V ,A) is a positive instance then|V (C)| 6 3k .

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Conflict Packing


Conflict Packing Lemma [Paul, P. and Thomasse, 2011]

Let T = (V ,A) be a tournament. There exists an ordering of T whosebackward arcs uv are such that u, v ∈ V (C).

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Conflict Packing


Lemma [Paul, P. and Thomasse, 2011]

Let T = (V ,A) be a tournament such that |V | > 4k . There exists a safepartition that can be computed in polynomial time.

proof

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Conflict Packing


Corollary [Paul, P. and Thomasse, 2011]

FEEDBACK ARC SET IN TOURNAMENTS admits a kernel with at most 4kvertices.

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Application to the RTI problem

Remove vertices that do not belong to any conflictSafe Partition reduction ruleConflict Packing allows to find a Safe Partition

Theorem [Paul, P. and Thomasse, 2011]

DENSE ROOTED TRIPLET INCONSISTENCY admits a kernel with at most5k vertices.

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Conclusion

6 Our results

7 Open problems

OUR RESULTS OPEN PROBLEMS

Main results

Polynomial kernelsFirst polynomial kernels:

(i) CLOSEST 3-LEAF POWER(ii) PROPER INTERVAL COMPLETION(iii) COGRAPH EDGE-EDITION

Improved polynomial kernels:

(i) FEEDBACK ARC SET IN TOURNAMENTS(ii) DENSE ROOTED TRIPLET INCONSISTENCY(iii) DENSE BETWEENNESS and DENSE CIRCULAR ORDERING

joint works with: S. Bessy, F. Fomin, S. Gaspers, S. Guillemot, F. Havet, C. Paul,S. Saurabh and S. Thomasse.

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Main results

Lower bounds on kernelization:(i) For any l > 7, the Pl -FREE EDGE-DELETION problem

does not admit a polynomial kernel.(ii) For any l > 4, the Cl -FREE EDGE-DELETION problem

does not admit a polynomial kernel.

joint work with: S. Guillemot, F. Havet and C. Paul.

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Open problems

Do the FEEDBACK VERTEX SET IN TOURNAMENTS and CLUSTER

VERTEX DELETION problems admit linear vertex-kernels?

Characterize lower bounds for modification problems. details

Can we use branches on other problems?(e.g. CHORDAL DELETION)Can we use Conflict Packing on other problems?(e.g. (weakly)-fragile constraint modification problems)

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Merci de votre attention !

kernelization algorithms for graph and other structure modiﬁcation problems

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