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Bart M. P. Jansen

Kernelization Lower Bounds

WorKer 2010, Leiden

Review of existing techniques and the introduction of cross-composition

Joint work with Hans L. Bodlaender and Stefan Kratsch

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Polynomial and Exponential Size Kernels

Some elusive FPT problems resisted all attempts to find polynomial kernels Connected Vertex Cover, k-Path, Treewidth, etc …

Existence of exponential-size kernels is implied by (uniform) fixed-parameter tractability

Tools to prove non-existence of polynomial kernels have been developed in recent years

Part I: Review of existing techniques for super-polynomial kernel lower bounds Emphasis on techniques Some applications as examples

Part II: Introducing cross-composition

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Outline

Part I Distillation algorithms OR-composition Poly-parameter

transformations

Part II

Cross composition

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PART I

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DISTILLATIONExisting techniques

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Weak distillation algorithms

Let A,B ⊆ S* be sets. A weak distillation of A into B is an algorithm which takes as input a sequence (x1, … , xt) of instances of A uses time polynomial in ∑i |xi| outputs x* with

x* ∈ B some xi ∈ A

|x*| is polynomial in maxi |xi|

If A = B then this is the notion of strong distillation (OR-distillation)

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poly(t*n) time

Weak distillation of A into B

x1 x2 x3 x4 x5 x6 x… xtA

instances

x*B

instance

n

poly(n)

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Consequences of weak distillation

Fortnow and Santhanam [STOC 2008] If set A is NP-hard under Karp reductions and there is a weak

distillation of A into any set B, then NP ⊆ coNP/poly Yap’s theorem [Theor. Comp. Sc. 1983]:

If NP ⊆ coNP/poly then the polynomial hierarchy collapses to the third level

Further collapses (Cai et al. [STACS 2003])

Intuitively: if 1 small instance of set B can express the logical OR of many

instances of the hard set A, then NP ⊆ coNP/poly small instance:

polynomial in size of largest input instance size independent of number of instances

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OR-COMPOSITIONExisting techniques

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Preliminaries

Given (x,k) ∈ S*×ℕ , its unparameterized version is the string: x#1111…1111 x#1k

If Q ⊆ S*×ℕ is a parameterized problem, then its unparameterized variant is Q := { x#1k | (x,k) ∈ Q }

1-to-1 correspondence between members of Q and Q

Parameter encoded in unary: polynomial-time transformation on an instance of Q yields polynomially-bounded blow-up in parameter size.

For a set A ⊆ S*, we define the set OR(A) as OR(A) := { (x1, x2, … , xt) | some xi ∈ A}

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OR-Composition

An OR-composition algorithm for a parameterized problem Q is an algorithm that takes as input a sequence (x1, k), (x2, k) , … , (xt, k) of

instances of Q with the same parameter value uses time polynomial in ∑i |xi| + k outputs (x*, k*) with

(x*, k*) ∈ Q some (xi, k) ∈ Q k* is polynomial in k

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poly(t*n + k) time

OR-composition of Q

Qinstance

Q instances x1 k x2 k x.. k xt k

n

x* k*

poly(k)

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Polynomial kernels for OR-compositional problems imply NP ⊆ coNP/poly

Bodlaender, Downey, Fellows, Hermelin: [ICALP 2008] If Q is a parameterized problem

which has a polynomial kernel which is OR-compositional whose unparameterized variant Q is NP-hard under Karp

reductions then there is a weak distillation from Q into OR(Q) and NP ⊆

coNP/poly*

Proof: we build a weak distillation algorithm from the given ingredients

* Refined statement and proof due to Holger Dell

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OR-composition + polynomial kernel Weak distillation of Q into OR(Q)

x1 x2 x3 x4 x5 x6 x… xtQ

instances

(x1,k1)Q instances

(x1,k2) (x1,k3) (x1,k4) (x1,k5) (x1,k6) (x…,k…) (xt,kt)

OR-Composed Q instances (y1,ki1) (y2,ki2) (y3,ki3) (yr,kir)

1 2 3 r

KernelizedQ

instances

(y’1,k’i1) (y’2,k’i2) (y’3,k’i3) (y’r,k’ir)

Q instances

x’1 x’2 x’3 x’r

Single OR(Q) instance (x’1, x’2 , x’3, x’r )

Parameterize

Group

Compose

Kernelize

Unparameterize

Tuple

Input

Output

n

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Application: OR-Composition for k-Path

Input: t instances of k-Path

Take disjoint union, output as (G’, k)

G’ has a k-path some Gi has a k-path Output parameter trivially bounded in poly(k)

,k ,k ,k ,k ,k

,k

k-Path does not admit a polynomial kernel unless

NP⊆coNP/poly

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POLYNOMIAL-PARAMETER TRANSFORMATIONS

Existing techniques

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Polynomial-parameter transformations

Let P and Q be parameterized problems A polynomial-parameter transformation from P to Q is an

algorithm which takes an instance (x,k) of P as input uses time polynomial in |x| + k outputs an instance (x’, k’) of Q with

(x,k) ∈ P (x’, k’) ∈ Q k’ is polynomial in k

Intuition: polynomial-time answer-preserving transformation of P to Q with bounded parameter increase

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Consequences of polynomial-parameter transformations

Bodlaender, Thomasse, Yeo: [ESA 2009] If there is a polynomial-parameter transformation from P to

Q and P and Q are NP-complete Q has a polynomial kernel

then P has a polynomial kernel

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Application of Polynomial-Parameter Transformations: Disjoint Cycles

Disjoint Cycles Input: Undirected simple graph G, integer k Parameter: k Question: Does G contain k vertex-disjoint simple cycles?

Goal: prove that Disjoint Cycles does not admit a polynomial kernel

Use polynomial-parameter transformations

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Proving a lower bound for Disjoint Cycles

MethodA. Introduce the NP-complete problem “Disjoint Factors”, prove it

does not have a polynomial kernel unless NP ⊆ coNP/polyB. Give a polynomial-parameter transformation from Disjoint

Factors to Disjoint Cycles

Reasoning Disjoint Cycles poly kernel Disjoint Factors poly kernel

(Theorem) No poly kernel for Disjoint Factors unless NP ⊆ coNP/poly Hence no poly kernel for Disjoint Cycles unless NP ⊆ coNP/poly

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A) Introducing Disjoint Factors

Disjoint Factors Input: Integer k, string S on alphabet {1, 2, … , k} Parameter: k Question: Can we find disjoint substrings S1, S2, … , Sk in S such

that Si starts and ends with i?

1432414132414231241214324141324142312412143241413241423124121432414132414231241214324141324142312412

Disjoint Factors does not admit a polynomial kernel

unless NP⊆coNP/poly

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B) Polynomial-parameter transformation

14324141324142312412

2 3 41

Input: Instance (S,k) of Disjoint Factors Output: Instance (G,k) of Disjoint Cycles String S has disjoint factorsG has k vertex-disjoint cycles

Disjoint Cycles does not admit a polynomial kernel

unless NP⊆coNP/poly

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Results through polynomial-parameter transformations

Incompressibility through colors and IDs Dom, Lokshtanov, Saurabh [ICALP 2009]

These problems do not have polynomial kernels unless NP ⊆ coNP/poly: Small Universe Set Cover

Parameter: |U| + k Small Universe Hitting Set

Parameter: |U| + k

Dominating Set parameterized by size of a vertex cover, Connected Vertex Cover, Steiner Tree, Small Subset Sum, etc.

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PART II

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THE MAIN IDEACross-composition

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Polynomial equivalence relationship

Let L be a set of strings R is a polynomial equivalence relationship on L if

R is an equivalence relationship R partitions any set of strings on at most n characters each

into poly(n) groups equivalency under R can be tested in polynomial time

Informally: an efficient way of grouping instances of size ≤n each into poly(n) groups

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Definition of cross-composition

Let L be a set of strings and Q a parameterized problem Set L cross-composes into Q if there is a polynomial

equivalence relationship R and an algorithm which takes as input t instances x1, … , xt of L which are equivalent

under R uses time polynomial in ∑i |xi| outputs an instance (x*, k*) of Q such that

(x*,k*) ∈ Q some xi ∈ L

k* is polynomial in maxi |xi| + log t

If set L cross-composes into parameterized problem Q: Then Q can express the OR of instances of L for a small

parameter value

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Comparison

OR-Composition An OR-composition for a

parameterized problem Q is an algorithm which

takes as input a sequence (x1, k), (x2, k) , … , (xt, k) of Q-instances

which share the same parameter

uses time polynomial in ∑i |xi| + k outputs (x*, k*) with

(x*, k*) ∈ Q some (xi, k) ∈ Q k* is polynomial in k

Cross-Composition A cross-composition of the set L

into parameterized problem Q is an algorithm which

takes as input a sequence x1, … , xt of L-instances

which are equivalent under some polynomial equivalence relationship

uses time polynomial in ∑i |xi| outputs (x*, k*) with

(x*,k*) ∈ Q some xi ∈ L,

k* is polynomial in maxi|xi|+log t

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Polynomial kernels for cross-compositional problems imply NP ⊆ coNP/poly

If there is a set A and parameterized problem Q such that set A is NP-hard under Karp reductions set A cross-composes into Q Q has a polynomial kernel

then there is a weak distillation from A into OR(Q) and NP⊆coNP/poly

Proof: We build a weak distillation

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Cross-composition + Polynomial kernel Weak distillation of A into OR(Q)

• In: t instances (x1, …, xt) of NP-hard set A• Define n := maxi |xi|

A) Input

• At most (|S|+1)n distinct inputs• Pairwise comparison to eliminate duplicates• Afterwards log t O(n)

B) Eliminate duplicates

• Partition inputs into groups X1, X2, … , Xr of inputs which are R-equivalent

• We get r poly(n) groups

C) Group by equivalence

• Cross-compose all inputs in group Xi into instance (xi*, ki*) of parameterized problem Q

• ki* is poly(n + log t), which is poly(n) since log t O(n)

D) Apply cross-composition

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Cross-composition + Polynomial kernel Weak distillation of A into OR(Q)

• Cross-compose all inputs in group Xi into instance (xi*, ki*) of parameterized problem Q

• ki* is poly(n + log t), which is poly(n) since log t O(n)

D) Apply cross-composition

• Kernelize each (xi*, ki*) to (xi’, ki’)• Afterwards |xi’|, ki’ ≤ poly(n)

E) Apply polynomial kernel for Q

• Transform (xi’, ki’) to unparameterized instance yi of Q• Size poly(n) per instance

F) Unparameterize

• Make tuple y* := (y1, y2, … , yr) which is an instance of OR(Q)• |y*| is r * poly(n)• |y*| is poly(n)

G) Build tuple: instance of OR(Q)

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AN APPLICATIONCross-composition

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Chromatic Number parameterized by Vertex Cover

Chromatic Number parameterized by Vertex Cover Input: Graph G, vertex cover Z of G, integer l. Parameter: k := |Z|. Question: Can the vertices of G be properly l -colored?

ZYES for l = 4

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Chromatic Number parameterized by Vertex Cover

Problem is FPT Simple

exponential-size kernel

No polynomial kernel unless NP ⊆ coNP/poly

Z

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Overview of the proof

Ingredients of the proofA. NP-completeness of 3-coloring on triangle split graphsB. Polynomial equivalence relationshipC. 3-coloring triangle split graphs cross-composes into Chromatic

Number parameterized by Vertex Cover

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A) Triangle split graphs

A triangle split graph is a graph G with vertex subset X: G[V – X] consists of vertex-disjoint triangles X is an independent set in G

V –X is a vertex cover

3-coloring is NP-complete on triangle split graphs

X

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B) Polynomial equivalence relationship

Two instances (G1, X1) and (G2, X2) of 3-coloring on triangle split graphs are equivalent under R if |V(G1)| = |V(G2)|, and |X1| = |X2|

Any set of instances on at most n vertices each is partitioned into n2 groups

R is a polynomial equivalence relationship

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χ(G…)≤3?χ(G1)≤3? χ(Gt)≤3?

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χ(G…)≤3?χ(G1)≤3? χ(Gt)≤3?

χ(G*)≤log t + 4?

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χ(G…)≤3?χ(G1)≤3? χ(Gt)≤3?

χ(G*)≤log t + 4?

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χ(G…)≤3?χ(G1)≤3? χ(Gt)≤3?

χ(G*)≤log t + 4?

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χ(G…)≤3?χ(G1)≤3? χ(Gt)≤3?

χ(G*)≤log t + 4?

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χ(G…)≤3?χ(G1)≤3? χ(Gt)≤3?

χ(G*)≤log t + 4?

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χ(G…)≤3?χ(G1)≤3? χ(Gt)≤3?

χ(G*)≤log t + 4?

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χ(G…)≤3?χ(G1)≤3? χ(Gt)≤3?

χ(G*)≤log t + 4?

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χ(G…)≤3?χ(G1)≤3? χ(Gt)≤3?

χ(G*)≤log t + 4?Klog t+4

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Conclusion of proof

For any fixed q, the q-Coloring problem parameterized by Vertex Cover does admit a polynomial kernel [BJK??]

Compare: 3-coloring parameterized by treewidth does not have a polynomial kernel (unless …) [BDFH ’08]

Chromatic Number par. by Vertex Cover does not admit a polynomial kernel unless

NP⊆coNP/poly

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CLIQUE PARAMETERIZED BY VERTEX COVER

Cross-composition

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Clique parameterized by Vertex Cover

Clique parameterized by Vertex Cover Input: Graph G, vertex cover Z of G, integer l. Parameter: k := |Z|. Question: Does G have a clique of size l?

ZYES for l = 5

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Clique parameterized by Vertex Cover

Problem is trivially FPT

Simple exponential-size kernel

Turing kernel: O(n) instances of |Z| + 1 vertices each

No polynomial kernel unless NP ⊆ coNP/poly

Z

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Cross-composing Clique into Clique parameterized by Vertex Cover

Input t instances (Gi, l) of unparameterized Clique, each looking for

an l-clique in a graph on n vertices

Output One instance (G’, l’, Z’) of Clique parameterized by Vertex

Cover, such that G’ has an l’-clique some Gi has an l-clique k’ = |Z’| is polynomial in n

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Vertex sets of G’

n

l

tInstance selectors

Vertex selectors

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Vertex sets of G’

n

l

2

n

tInstance selectors

Vertex selectors

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Vertex sets of G’

n

l

2

n

tInstance selectors

Vertex selectors

Edge checkers

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The vertex cover

n

l

2

n

tInstance selectors

Vertex selectors

Edge checkers

Vertex cover

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Conclusion of proof

Strengthens result of [BDFH ‘08] that Clique parameterized by Treewidth does not have a polynomial kernel

Clique par. by Vertex Cover does not admit a polynomial kernel unless NP⊆coNP/poly

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THE BIGGER PICTURECross-composition

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Advantages of cross-composition

• No need for problem-specific padding arguments

Polynomial equivalence relationship

• No need for single-exponential FPT algorithm

Output parameter may depend on log t

• Facilitates the encoding of input instances at bounded parameter cost

Output parameter may depend on maxi |xi|

• Starting from a restricted version of the problem makes the input instances well-behaved

Start from any NP-hard problem

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Cross-composition unifies existing techniques

OR-composition OR-composition of Q Unparameterized variant Q

cross-composes into Q

Poly-param. transforms OR-composition of P and

polynomial-parameter transformation P Q

Unparameterized variant P cross-composes into Q

Both existing techniques for kernel lower bounds actually prove that there is a cross-composition

Intuition: parameterized problem Q does not admit a polynomial kernel if it can express the OR of some NP-hard problem at small parameter cost

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Conclusions and discussion

Techniques for proving conditional kernel lower-bounds: show that polynomial kernel weak distillation

AND-composition Prove kernel lower bounds based on a conjecture; no interesting

consequences known if this conjecture fails Treewidth, Cliquewidth, (…)-width do not have polynomial kernels

unless this conjecture fails Cross-composition relaxes the requirements and hence

simplifies the proofs of lower bounds Clique and Chromatic Number parameterized by the size of a

Vertex Cover do not admit polynomial kernels unless NP ⊆ coNP/poly

Future work: prove kernel lower bounds for more problems! Edge Clique Cover H-Minor-free Deletion

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List of FPT problems without polynomial kernels unless NP ⊆ coNP/poly

[HN06+FS08] k-Variable CNF-SAT [BDFH08] Longest Path, Longest Cycle [BTY09] Vertex Disjoint Paths, Cycles [DLS09] Bounded Universe Hitting Set, Bounded Universe Set Cover,

Connected Vertex Cover, Steiner Tree, Capacitated Vertex Cover [KW09] Windmill-free Edge-Deletion [KW09’] Cases of MinOnesSat [FJLRS10] Dogson Score [CPPW10] Connectivity problems in d-degenerate graphs: Connected

Feedback Vertex Set, Connected Dominating Set, Connected Odd Cycle Transversal

[KMW10] MaxOnesSat and ExactOnesSat [BJ??] Weighted Vertex Cover parameterized by P2-deletion distance [BJK??] Clique parameterized by Vertex Cover, Chromatic Number

parameterized by Vertex Cover, non-standard parameterizations of Feedback Vertex Set

[FFPS11] Total Vertex (Edge) Cover

Thank you!

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Part I Distillation OR-composition Poly-parameter transformations

Part II Cross-composition Chromatic Number Clique