A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Abstract

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions.

On Lower Bounds of Approximating Parameterized $k$-Clique

“…We state it in terms of inapproximability of DENSEST k-SUBGRAPH as follows. Note that the original statement of PIH in [LRSZ20] says this problem is W[1]-hard to approximate, but for our use, we choose a relaxed form which states that it has no constant ratio FPT-approximation algorithm, as in [FKLM20].…”

“…Thus, PIH implies k-CLIQUE does not admit constant ratio FPT-approximation algorithms. However, the other direction is not necessarily true (forbidding small clique does not imply low edge density), and it remains an important open problem that whether PIH holds if we assume k-CLIQUE is hard to approximate within any constant factor in FPT time [FKLM20].…”

On Hardness of Approximation of Parameterized Set Cover and Label Cover: Threshold Graphs from Error Correcting Codes

“…In these results the gap is inherent in the assumption, and the challenge is to construct gap-preserving reductions. These results are not the focus of this paper and we shall not elaborate further on them, and the interested reader may see the recent survey of Feldman et al [FKLM20] for more details.…”

Strong Hardness of Approximation for Tree Transversals

“…Note that in this paper, we focus on the notion of subgraphs instead of induced subgraphs and (topological) minors, both of which have been actively studied through the lens of approximation and parameterized algorithms. We refer the reader to recent papers [AKL20, FLP + 20, KLT21] and a survey [FKLM20] on these topics.…”