Kernelization Techniques and Cross-Composition for FPT Problems

Bart M. P. JansenKernelization Lower Bounds Review of existing techniques and the introduction of cross-composition Joint work with Hans L. Bodlaender and Stefan Kratsch WorKer 2010, Leiden

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

Outline Part I Part II Cross composition • Distillation algorithms • OR-composition • Poly-parameter transformations

part I

Existing techniques Distillation

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)

Weak distillation of A into B poly(t*n) time A instances x1 x2 x3 x4 x5 x6 x… xt poly(n) n B instance x*

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

Existing techniques or-composition

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 Qyields • 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}

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

OR-composition of Q poly(t*n + k) time Qinstances x1 k x2 k x.. k xt k poly(k) n Q instance x* k*

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

OR-composition + polynomial kernel  Weak distillation of Q into OR(Q) Qinstances x1 x2 x3 x4 x5 x6 x… xt Unparameterize Parameterize Compose Input Kernelize Group Output Tuple n Qinstances (x1,k1) (x1,k2) (x1,k3) (x1,k4) (x1,k5) (x1,k6) (x…,k…) (xt,kt) 1 2 3 r OR-Composed Q instances (y1,ki1) (y2,ki2) (y3,ki3) (yr,kir) 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 )

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

Existing techniques polynomial-parameter transformations

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

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

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

Proving a lower bound for Disjoint Cycles • Method • Introduce the NP-complete problem “Disjoint Factors”, prove it does not have a polynomial kernel unless NP ⊆ coNP/poly • 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

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? 14324141324142312412 14324141324142312412 14324141324142312412 14324141324142312412 14324141324142312412 Disjoint Factors does not admit a polynomial kernel unless NP⊆coNP/poly

B) Polynomial-parameter transformation • Input: Instance (S,k) of Disjoint Factors • Output: Instance (G,k) of Disjoint Cycles • String S has disjoint factorsG has k vertex-disjoint cycles 14324141324142312412 1 2 3 4 Disjoint Cycles does not admit a polynomial kernel unless NP⊆coNP/poly

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.

part II

Cross-composition the main idea

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

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

Comparison OR-Composition 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 • 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

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

Cross-composition + Polynomial kernel  Weak distillation of A into OR(Q) A) Input • In: t instances (x1, …, xt) of NP-hard set A • Define n := maxi |xi| B) Eliminate duplicates • At most (|S|+1)n distinct inputs • Pairwise comparison to eliminate duplicates • Afterwards log t  O(n) C) Group by equivalence • Partition inputs into groups X1, X2, … , Xr of inputs which are R-equivalent • We get r  poly(n) groups D) Apply cross-composition • 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)

Cross-composition + Polynomial kernel  Weak distillation of A into OR(Q) D) Apply cross-composition • 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) E) Apply polynomial kernel for Q • Kernelize each (xi*, ki*) to (xi’, ki’) • Afterwards |xi’|, ki’ ≤ poly(n) F) Unparameterize • Transform (xi’, ki’) to unparameterized instance yi of Q • Size poly(n) per instance G) Build tuple: instance of OR(Q) • Make tuple y* := (y1, y2, … , yr) which is an instance of OR(Q) • |y*| is r * poly(n) • |y*| is poly(n)

Cross-composition An application

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? Z YES for l = 4

Chromatic Number parameterized by Vertex Cover • Problem is FPT • Simple exponential-size kernel • No polynomial kernel unless NP ⊆ coNP/poly Z

Overview of the proof • Ingredients of the proof • NP-completeness of 3-coloring on triangle split graphs • Polynomial equivalence relationship • 3-coloring triangle split graphs cross-composes into Chromatic Number parameterized by Vertex Cover

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

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

χ(G1)≤3? χ(G…)≤3? χ(Gt)≤3?

χ(G1)≤3? χ(G…)≤3? χ(Gt)≤3? χ(G*)≤log t + 4?

χ(G1)≤3? χ(G…)≤3? χ(Gt)≤3? Klog t+4 χ(G*)≤log t + 4?

Conclusion of proof Chromatic Number par. by Vertex Cover does not admit a polynomial kernel unless NP⊆coNP/poly • 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]

Cross-composition Clique parameterized by vertex cover

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? Z YES for l = 5

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

Kernelization Techniques and Cross-Composition for FPT Problems