Fractals for Kernelization Lower Bounds

Abstract

The composition technique is a popular method for excluding polynomial-size problem kernels for NP-hard parameterized problems. We present a new technique exploiting triangle-based fractal structures for extending the range of applicability of compositions. Our technique makes it possible to prove new no-polynomial-kernel results for a number of problems dealing with length-bounded cuts. In particular, answering an open question of Golovach and Thilikos [Discrete Optim. 2011], we show that, unless NP $\subseteq$ coNP / poly, the NP-hard Length-Bounded Edge-Cut (LBEC) problem (delete at most $k$ edges such that the resulting graph has no $s$-$t$ path of length shorter than $\ell$) parameterized by the combination of $k$ and $\ell$ has no polynomial-size problem kernel. Our framework applies to planar as well as directed variants of the basic problems and also applies to both edge and vertex deletion problems. Along the way, we show that LBEC remains NP-hard on planar graphs, a result which we believe is interesting in its own right.