Diminishable Parameterized Problems and Strict Polynomial Kernelization

TL;DR

This paper introduces the concept of diminishable problems and demonstrates that many fixed-parameter tractable problems do not admit strict polynomial kernels unless P=NP, highlighting limitations in kernelization techniques.

Contribution

It develops a simple framework showing the non-existence of strict polynomial kernels for various problems under P≠NP, using the concept of diminishable problems.

Findings

Many FPT problems lack strict polynomial kernels unless P=NP.

Diminishable problems can reduce parameters polynomially while preserving problem equivalence.

Relaxations of strict kernels have significant limitations.

Abstract

Kernelization---a mathematical key concept for provably effective polynomial-time preprocessing of NP-hard problems---plays a central role in parameterized complexity and has triggered an extensive line of research. This is in part due to a lower bounds framework that allows to exclude polynomial-size kernels under the assumption of NP $\nsubseteq$ coNP$/$poly. In this paper we consider a restricted yet natural variant of kernelization, namely strict kernelization, where one is not allowed to increase the parameter of the reduced instance (the kernel) by more than an additive constant. Building on earlier work of Chen, Flum, and M\"{u}ller [Theory Comput. Syst. 2011] and developing a general and remarkably simple framework, we show that a variety of FPT problems does not admit strict polynomial kernels under the weaker assumption of P $\neq$ NP. In particular, we show that various (multicolored) graph problems and Turing machine computation problems do not admit strict polynomial kernels unless P $=$ NP. To this end, a key concept we use are diminishable problems; these are parameterized problems that allow to decrease the parameter of the input instance by at least one in polynomial time, thereby outputting an equivalent problem instance. Finally, we study a relaxation of the notion of strict kernels and reveal its limitations.