Kernelization Lower Bounds Through Colors and IDs

@article{Dom2014KernelizationLB,
  title={Kernelization Lower Bounds Through Colors and IDs},
  author={Michael Dom and Daniel Lokshtanov and Saket Saurabh},
  journal={ACM Transactions on Algorithms (TALG)},
  year={2014},
  volume={11},
  pages={1 - 20}
}
In parameterized complexity, each problem instance comes with a parameter k, and a parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance to an instance with size polynomial in k. Many problems have been shown to admit polynomial kernels, but it is only recently that a framework for showing the nonexistence of polynomial kernels for specific problems has been developed by Bodlaender et al. [2009] and Fortnow and… 

Figures from this paper

Kernelization of packing problems

This work shows lower bounds for the kernelization of d-Set Matching and other packing problems, and applies this scheme to the vertex cover problem, which allows us to replace the number-theoretical construction by Dell and Van Melkebeek with shorter elementary arguments.

Diminishable Parameterized Problems and Strict Polynomial Kernelization

This paper shows that a variety of FPT problems does not admit strict polynomial kernels under the weaker assumption of P $\neq$ NP, and uses diminishable problems, which are parameterized problems that allow to decrease the parameter of the input instance by at least one in polynometric time, thereby outputting an equivalent problem instance.

Tight Kernel Bounds for Problems on Graphs with Small Degeneracy

This article considers kernelization for problems on d-degenerate graphs, that is, graphs such that any subgraph contains a vertex of degree at most d, and proves that unless coNP ⊆ NP/poly Dominating Set has no kernels of size O(kd−1)(d−3)−ε) for any ε > 0.

A ug 2 01 7 Diminishable Parameterized Problems and Strict Polynomial Kernelization

This paper shows that various (multicolored) graph problems and Turing machine computation problems do not admit strict polynomial kernels unless P = NP, and studies a relaxation of the notion of strict kernels and reveal its limitations.

Optimal Sparsification for Some Binary CSPs Using Low- Degree Polynomials

This paper analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems, without changing the answer, and characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments.

Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials

This article analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems without changing the answer, and characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments.

On polynomial kernels for sparse integer linear programs

Sparsification Upper and Lower Bounds for Graph Problems and Not-All-Equal SAT

It is shown that for a number of decision problems on graphs, polynomial-time algorithms cannot compress instances of such problems to equivalent instances, of a possibly different problem, whose encoding size is sub-quadratic in the number of vertices.

On the Approximate Compressibility of Connected Vertex Cover

This paper considers parameters that are strictly smaller than the size of the solution and obtains the first polynomial size approximate kernelization schemes for the Connected Vertex Cover problem when parameterized by the deletion distance of the input graph.

How Much Does a Treedepth Modulator Help to Obtain Polynomial Kernels Beyond Sparse Graphs?

This article proves that Vertex Cover admits a polynomial kernel on general graphs for any integer c, and that Dominating Set does not for anyinteger $$c \ge 2$$c≥2 even on degenerate graphs, unless $$\text {NP} \subseteq \text {coNP}/\text{poly}$$NP⊆coNP/poly.
...

References

SHOWING 1-10 OF 36 REFERENCES

Cross-Composition: A New Technique for Kernelization Lower Bounds

It is shown that if an NP-hard problem cross-composes into a parameterized problem Q then Q does not admit a polynomial kernel unless thePolynomial hierarchy collapses, and its applicability is shown by proving kernelization lower bounds for a number of important graphs problems with structural (non-standard) parameterizations.

Co-Nondeterminism in Compositions: A Kernelization Lower Bound for a Ramsey-Type Problem

This work presents the first example of how co-nondeterminism can help to make a composition algorithm, and studies the existence of polynomial kernels for a Ramsey-type problem where, given a graph G and an integer k, the question is whether G contains an independent set or a clique of size at least k.

The Parameterized Complexity of the Unique Coverage Problem

This work considers the parameterized complexity of the UNIQUE COVERAGE problem: given a family of sets and a parameter k, whether there exists a subfamily that covers at least k elements exactly once, and shows that this problem is fixed-parameter tractable with respect to the parameter k.

Kernels for the Dominating Set Problem on Graphs with an Excluded Minor

The results imply that there is a problem kernel of polynomial size for graphs with an excluded minor and a linear kernel for graphs that are K3,h-minor-free.

Kernel bounds for disjoint cycles and disjoint paths

Evidence is given that DisJoint Cycles and Disjoint Paths do not have polynomial kernels, unless NP ⊆ coNP/poly, and it is shown that the related Disj Joint Cycles Packing problem has a kernel of size O(k logk).

On problems without polynomial kernels

Capacitated Domination and Covering: A Parameterized Perspective

Capacitated Vertex Cover is the first known "subset problem" which has turned out to be fixed parameter tractable when parameterized by solution size butW[1]-hard when parameterizing by treewidth.

Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses

It is shown that if satisfiability for n-variable d-CNF formulas has a protocol of cost O(nd-ε) then coNP is in NP/poly, which implies that the polynomial-time hierarchy collapses to its third level.

Kernel(s) for problems with no kernel: On out-trees with many leaves

This work gives the first polynomial kernel for Rooted k-Leaf-Out-Branching, a variant of k- leaf- out-branching where the root of the tree searched for is also a part of the input, and is the first ones separating Karp kernelization from Turing kernelization.