Cross-Composition: A New Technique for Kernelization Lower Bounds

Hans L. Bodlaender (1) , Bart Jansen (1) , Stefan Kratsch (1)

1 Department of Information and Computing Sciences [Utrecht]

We introduce a new technique for proving kernelization lower bounds, called \emph{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-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 \orsymb-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., \textsc{Chromatic Number}, \textsc{Clique}, and \textsc{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 \textsc{Feedback Vertex Set}.

Mots clés: kernelization; lower bounds; parameterized complexity

Domaines: Complexité [cs.CC]; Algorithme et structure de données [cs.DS]

Hans L. Bodlaender, Bart Jansen, Stefan Kratsch. Cross-Composition: A New Technique for Kernelization Lower Bounds. Symposium on Theoretical Aspects of Computer Science (STACS2011), Mar 2011, Dortmund, Germany. pp.165-176. ⟨hal-00573603⟩