Stefan Kratsch

Cross-Composition: A New Technique for Kernelization Lower Bounds

We introduce a new technique for proving kernelization lower bounds, called cross-composition. A classical problem L cross-composes into a parameterized problem

Kernelization Lower Bounds by Cross-Composition

Q

Representative Sets and Irrelevant Vertices: New Tools for Kernelization

if an instance of Q with polynomially bounded parameter value can express the logical OR of a sequence of instances of L. Building on work by Bodlaender et al. (ICALP 2008) and using a result by Fortnow and Santhanam (STOC 2008) we show that if an NP-hard problem cross-composes into a parameterized problem Q then Q does not admit a polynomial kernel unless the polynomial hierarchy collapses. Our technique generalizes and strengthens the recent techniques of using OR-composition algorithms and of transferring the lower bounds via polynomial parameter transformations. We show its applicability by proving kernelization lower bounds for a number of important graphs problems with structural (non-standard) parameterizations, e.g., Chromatic Number, Clique, and Weighted Feedback Vertex Set do not admit polynomial kernels with respect to the vertex cover number of the input graphs unless the polynomial hierarchy collapses, contrasting the fact that these problems are trivially fixed-parameter tractable for this parameter. We have similar lower bounds for Feedback Vertex Set.

Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal

We introduce the framework of cross-composition for proving kernelization lower bounds. A classical problem

Preprocessing for Treewidth: A Combinatorial Analysis through Kernelization

L

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

\and/or-cross-composes into a parameterized problem

Two Edge Modification Problems without Polynomial Kernels

\mathcal{Q}

Clique Cover and Graph Separation: New Incompressibility Results

if it is possible to efficiently construct an instance of

Preprocessing for treewidth: a combinatorial analysis through kernelization

\mathcal{Q}

Isomorphism for graphs of bounded feedback vertex set number

with polynomially bounded parameter value that expresses the logical and or or of a sequence of instances of