Parameterized (in)approximability of subset problems

TL;DR

This paper explores the limits of approximability and inapproximability for a broad class of subset problems within fixed-parameter tractability, introducing new concepts and results on parameterized approximability.

Contribution

It introduces intersective approximability, generalizes safe approximability, and provides strong inapproximability and approximation results for various subset problems under different parameterizations.

Findings

Many problems are W[\cdot]-hard but admit parameterized approximation schemes.

Introduces the notion of intersective approximability for subset problems.

Shows inapproximability results for several well-known problems.

Abstract

We discuss approximability and inapproximability in FPT-time for a large class of subset problems where a feasible solution $S$ is a subset of the input data and the value of $S$ is $|S|$. The class handled encompasses many well-known graph, set, or satisfiability problems such as Dominating Set, Vertex Cover, Set Cover, Independent Set, Feedback Vertex Set, etc. In a first time, we introduce the notion of intersective approximability that generalizes the one of safe approximability and show strong parameterized inapproximability results for many of the subset problems handled. Then, we study approximability of these problems with respect to the dual parameter $n-k$ where $n$ is the size of the instance and $k$ the standard parameter. More precisely, we show that under such a parameterization, many of these problems, while W[$\cdot$]-hard, admit parameterized approximation schemata.