Clique cover and graph separation: New incompressibility results

TL;DR

This paper establishes new lower bounds on the kernelization complexity of several graph problems, showing they do not admit polynomial kernels unless a major complexity-theoretic collapse occurs.

Contribution

It proves that key graph problems like EDGE CLIQUE COVER and MULTICUT do not have polynomial kernels under standard complexity assumptions, resolving open problems in kernelization.

Findings

Proves no polynomial kernel for EDGE CLIQUE COVER unless NP⊆coNP/poly.

Shows no polynomial kernel for MULTICUT problems under standard assumptions.

Complements recent algorithmic advances in cut problems.

Abstract

The field of kernelization studies polynomial-time preprocessing routines for hard problems in the framework of parameterized complexity. Although a framework for proving kernelization lower bounds has been discovered in 2008 and successfully applied multiple times over the last three years, establishing kernelization complexity of many important problems remains open. In this paper we show that, unless NP is a subset of coNP/poly and the polynomial hierarchy collapses up to its third level, the following parameterized problems do not admit a polynomial-time preprocessing algorithm that reduces the size of an instance to polynomial in the parameter: - EDGE CLIQUE COVER, parameterized by the number of cliques, - DIRECTED EDGE/VERTEX MULTIWAY CUT, parameterized by the size of the cutset, even in the case of two terminals, - EDGE/VERTEX MULTICUT, parameterized by the size of the cutset, and - k-WAY CUT, parameterized by the size of the cutset. The existence of a polynomial kernelization for EDGE CLIQUE COVER was a seasoned veteran in open problem sessions. Furthermore, our results complement very recent developments in designing parameterized algorithms for cut problems by Marx and Razgon [STOC'11], Bousquet et al. [STOC'11], Kawarabayashi and Thorup [FOCS'11] and Chitnis et al. [SODA'12].