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### 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

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)

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

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

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

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 factorsG 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.

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)

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

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]

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

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

Conclusion of proof

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

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

Advantages of cross-composition

- Polynomial equivalence relationship

- No need for problem-specific padding arguments

- Output parameter may depend on log t

- No need for single-exponential FPT algorithm

- Output parameter may depend on maxi |xi|

- Facilitates the encoding of input instances at bounded parameter cost

- Start from any NP-hard problem

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

Cross-composition unifies existing techniques

OR-composition

Poly-param. transforms

OR-composition of P and polynomial-parameter transformation P Q

Unparameterized variant P cross-composes into Q

- OR-composition of Q
- Unparameterized variant Q 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

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

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!

- Distillation
- OR-composition
- Poly-parameter transformations
- Part II
- Cross-composition
- Chromatic Number
- Clique

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