A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Feldmann, Andreas Emil; Karthik, C.; Lee, Euiwoong; Manurangsi, Pasin

doi:10.3390/a13060146

Cited by 39 publications

(23 citation statements)

References 253 publications

(359 reference statements)

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“…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].…”

Section: Hypothesesmentioning

confidence: 99%

“…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].…”

Section: Hypothesis 5 (Parameterized Inapproximability Hypothesis (Pi...mentioning

confidence: 99%

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On Lower Bounds of Approximating Parameterized $k$-Clique

Lin

Ren

Sun

et al. 2021

Preprint

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Given a simple graph G and an integer k, the goal of k-CLIQUE problem is to decide if G contains a complete subgraph of size k. We say an algorithm approximates k-CLIQUE within a factor g(k) if it can find a clique of size at least k/g(k) when G is guaranteed to have a kclique. Recently, it was shown that approximating k-CLIQUE within a constant factor is WWe study the approximation of k-CLIQUE under the Exponential Time Hypothesis (ETH). The reduction of [Lin21] already implies an n Ω( 6 √ log k) -time lower bound under ETH. We improve this lower bound to n Ω(log k) . Using the gap-amplification technique by expander graphs, we also prove that there is no k o(1) factor FPT-approximation algorithm for k-CLIQUE under ETH.We also suggest a new way to prove the Parameterized Inapproximability Hypothesis (PIH) under ETH. We show that if there is no n O( k log k ) algorithm to approximate k-CLIQUE within a constant factor, then PIH is true.

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Section: Hypothesesmentioning

confidence: 99%

Section: Hypothesis 5 (Parameterized Inapproximability Hypothesis (Pi...mentioning

confidence: 99%

On Lower Bounds of Approximating Parameterized $k$-Clique

Lin

Ren

Sun

et al. 2021

Preprint

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“…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.…”

Section: Introductionmentioning

confidence: 96%

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

Karthik¹,

Navon²

2021

2021 Proceedings of the Workshop on Algorithm Engineering and Experiments (ALENEX)

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Strong Hardness of Approximation for Tree Transversals

Lee¹,

Wang²

2021

Preprint

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Let H be a fixed graph. The H-Transversal problem, given a graph G, asks to remove the smallest number of vertices from G so that G does not contain H as a subgraph. While a simple |V (H)|-approximation algorithm exists and is believed to be tight for every 2-vertex-connected H, the best hardness of approximation for any tree was Ω(log |V (H)|)-inapproximability when H is a star.In this paper, we identify a natural parameter ∆ for every tree T and show that T -Transversal is NP-hard to approximate within a factor (∆ − 1 − ε) for an arbitrarily small constant ε > 0. As a corollary, we prove that there exists a tree T such that T -Transversal is NP-hard to approximate within a factor Ω(|V (T )|), exponentially improving the best known hardness of approximation for tree transversals.

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A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Cited by 39 publications

References 253 publications

On Lower Bounds of Approximating Parameterized $k$-Clique

On Lower Bounds of Approximating Parameterized $k$-Clique

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

Strong Hardness of Approximation for Tree Transversals

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