Fractals for Kernelization Lower Bounds

TL;DR

This paper introduces a fractal-based composition technique to establish new lower bounds on polynomial kernels for length-bounded cut problems, resolving an open question and extending kernelization theory.

Contribution

It presents a novel fractal structure technique for kernelization lower bounds, applicable to various graph problems including planar and directed variants.

Findings

Proves no polynomial kernel for Length-Bounded Edge-Cut unless NP ⊆ coNP/poly.

Extends lower bounds to planar and directed graph variants.

Shows LBEC remains NP-hard on planar graphs.

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.