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Bart M. P. Jansen Kernelization 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.

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bart m p jansen kernelization lower bounds

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
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
Outline

Part I

Part II

Cross composition

  • Distillation algorithms
  • OR-composition
  • Poly-parameter transformations
weak distillation algorithms
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
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
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
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 composition1
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
OR-composition of Q

poly(t*n + k) time

Qinstances

x1

k

x2

k

x..

k

xt

k

poly(k)

n

Q

instance

k*

x*

polynomial kernels for or compositional problems imply np conp poly
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
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
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 transformations1
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
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
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
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
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
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
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
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
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
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
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
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 q1
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
  • 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 cover1
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
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 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
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
slide38

χ(G1)≤3?

χ(G…)≤3?

χ(Gt)≤3?

slide39

χ(G1)≤3?

χ(G…)≤3?

χ(Gt)≤3?

χ(G*)≤log t + 4?

slide40

χ(G1)≤3?

χ(G…)≤3?

χ(Gt)≤3?

χ(G*)≤log t + 4?

slide41

χ(G1)≤3?

χ(G…)≤3?

χ(Gt)≤3?

χ(G*)≤log t + 4?

slide42

χ(G1)≤3?

χ(G…)≤3?

χ(Gt)≤3?

χ(G*)≤log t + 4?

slide43

χ(G1)≤3?

χ(G…)≤3?

χ(Gt)≤3?

χ(G*)≤log t + 4?

slide44

χ(G1)≤3?

χ(G…)≤3?

χ(Gt)≤3?

χ(G*)≤log t + 4?

slide45

χ(G1)≤3?

χ(G…)≤3?

χ(Gt)≤3?

χ(G*)≤log t + 4?

slide46

χ(G1)≤3?

χ(G…)≤3?

χ(Gt)≤3?

Klog t+4

χ(G*)≤log t + 4?

conclusion of proof
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 cover1
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 cover2
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
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
vertex sets of g
Vertex sets of G’

Instance selectors

t

l

n

Vertex selectors

vertex sets of g1
Vertex sets of G’

Instance selectors

t

l

n

Vertex selectors

vertex sets of g2
Vertex sets of G’

Instance selectors

t

l

n

Vertex selectors

Edge checkers

the vertex cover
The vertex cover

Instance selectors

t

Vertex cover

l

n

Vertex selectors

Edge checkers

conclusion of proof1
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
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
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
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
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!

slide62

Part I

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