On problems without polynomial kernels

Abstract

Kernelization is a strong and widely-applied technique in parameterized complexity. A kernelization algorithm, or simply a kernel, is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parameter bounded by a function of the parameter in the input. A kernel is polynomial if the size and parameter of the output are polynomially-bounded by the parameter of the input. In this paper we develop a framework which allows showing that a wide range of FPT problems do not have polynomial kernels. Our evidence relies on hypothesis made in the classical world (i.e. non-parametric complexity), and revolves around a new type of algorithm for classical decision problems, called a distillation algorithm, which is of independent interest. Using the notion of distillation algorithms, we develop a generic lower-bound engine that allows us to show that a variety of FPT problems, fulfilling certain criteria, cannot have polynomial kernels unless the polynomial hierarchy collapses. These problems include k-Path, k-Cycle, k-Exact Cycle, k-Short Cheap Tour, k-Graph Minor Order Test, k-Cutwidth, k-Search Number, k-Pathwidth, k-Treewidth, k-Branchwidth, and several optimization problems parameterized by treewidth and other structural parameters.

A Multivariate Complexity Analysis of Lobbying in Multiple Referenda

“…The corresponding kernels, however, may have exponential size and it is of particular interest to determine which problems, with respect to which parameter(s), allow for polynomial-size problem kernels (Bodlaender, 2009;Guo & Niedermeier, 2007) since the total running time complexity may vary dependent on the kernel size. Using techniques developed by Bodlaender, Downey, Fellows, and Hermelin (2009) and by Fortnow andSanthanam (2011), Szeider (2011) recently proposed to examine the power of kernelization for several problems in Artificial Intelligence. See Section 4 for more discussion.…”

Imbalance Is Fixed Parameter Tractable

“…This is achieved in 2 It is well-known that every problem that is FPT admits a kernel (see [15] for definition). While graph layout problems such as Treewidth, Pathwidth and Cutwidth are FPT [3,5,17] one can easily show using recently developed machinery [4] that they are unlikely to admit polynomial kernels. Giving such a lower bound for Imbalance seems non-trivial, while a polynomial kernel for the problem would be the first such kernel for a graph layout problem.…”

Parameterized Complexity of Arc-Weighted Directed Steiner Problems

“…We show the claim by applying the lower bound technique of Bodlaender et al [5] to DST. More specifically, we show that there exists a composition algorithm for DST, which implies, according to Lemmas 1 and 2 of [5], that a polynomial problem kernel for DST with respect to the combined parameter would lead to NP ⊆ coNP∕ poly.…”