FPT Approximations for Packing and Covering Problems Parameterized by Elimination Distance and Even Less

Abstract

For numerous graph problems in the realm of parameterized algorithms, using the size of a smallest deletion set (called a modulator) into well-understood graph families as parameterization has led to a long and successful line of research. Recently, however, there has been an extensive study of structural parameters that are potentially much smaller than the modulator size. In particular, recent papers [Jansen et al. STOC 2021; Agrawal et al. SODA 2022] have studied parameterization by the size of the modulator to a graph family ℋ(mod_ℋ(⋅)), elimination distance to ℋ(ed_ℋ(⋅)), and ℋ-treewidth (tw_ℋ(⋅)). These parameters are related by the fact that tw_ℋ lower bounds ed_ℋ, which in turn lower bounds mod_ℋ. While these new parameters have been successfully exploited to design fast exact algorithms their utility (especially that of ed_ℋ and tw_ℋ) in the context of approximation algorithms is mostly unexplored. 
The conceptual contribution of this paper is to present novel algorithmic meta-theorems that expand the impact of these structural parameters to the area of FPT Approximation, mirroring their utility in the design of exact FPT algorithms. Precisely, we show that if a covering or packing problem is definable in Monadic Second Order Logic and has a property called Finite Integer Index (FII), then the existence of an FPT Approximation Scheme (FPT-AS, i.e., (1±ε)-approximation) parameterized by mod_ℋ(⋅), ed_ℋ(⋅), and tw_ℋ(⋅) is in fact equivalent. As a consequence, we obtain FPT-ASes for a wide range of covering, packing, and domination problems on graphs with respect to these parameters. In the process, we show that several graph problems, that are W[1]-hard parameterized by mod_ℋ, admit FPT-ASes not only when parameterized by mod_ℋ, but even when parameterized by the potentially much smaller parameter tw_ℋ(⋅). In the spirit of [Agrawal et al. SODA 2022], our algorithmic results highlight a broader connection between these parameters in the world of approximation. As concrete exemplifications of our meta-theorems, we obtain FPT-ASes for well-studied graph problems such as Vertex Cover, Feedback Vertex Set, Cycle Packing and Dominating Set, parameterized by tw_ℋ(⋅) (and hence, also by mod_ℋ(⋅) or ed_ℋ(⋅)), where ℋ is any family of minor free graphs.