Cross-Composition: A New Technique for Kernelization Lower Bounds

TL;DR

This paper introduces cross-composition, a novel technique for establishing kernelization lower bounds, demonstrating that several graph problems lack polynomial kernels under standard complexity assumptions.

Contribution

The paper presents the cross-composition technique, extending previous methods to prove kernelization lower bounds for various graph problems with structural parameters.

Findings

Chromatic Number, Clique, and Weighted Feedback Vertex Set lack polynomial kernels unless the polynomial hierarchy collapses.

The technique generalizes and strengthens existing OR-composition methods.

Applied to multiple problems, showing limitations of fixed-parameter tractability.

Abstract

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