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# Hardness of Approximating Bounded-Degree Max 2-CSP and Independent Set on k-Claw-Free Graphs

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LIPIcs.ITCS.2024.71.pdf
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## Acknowledgements

This work was initiated at Dagstuhl Seminar 23291 "Parameterized Approximation: Algorithms and Hardness". We thank the organizers and participants of the workshop for helpful discussions.

## Cite As

Euiwoong Lee and Pasin Manurangsi. Hardness of Approximating Bounded-Degree Max 2-CSP and Independent Set on k-Claw-Free Graphs. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 71:1-71:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.71

## Abstract

We consider the question of approximating Max 2-CSP where each variable appears in at most d constraints (but with possibly arbitrarily large alphabet). There is a simple ((d+1)/2)-approximation algorithm for the problem. We prove the following results for any sufficiently large d: - Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized reduction) to approximate this problem to within a factor of (d/2 - o(d)). - It is NP-hard (under randomized reduction) to approximate the problem to within a factor of (d/3 - o(d)). Thanks to a known connection [Pavel Dvorák et al., 2023], we establish the following hardness results for approximating Maximum Independent Set on k-claw-free graphs: - Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized reduction) to approximate this problem to within a factor of (k/4 - o(k)). - It is NP-hard (under randomized reduction) to approximate the problem to within a factor of (k/(3 + 2√2) - o(k)) ≥ (k/(5.829) - o(k)). In comparison, known approximation algorithms achieve (k/2 - o(k))-approximation in polynomial time [Meike Neuwohner, 2021; Theophile Thiery and Justin Ward, 2023] and (k/3 + o(k))-approximation in quasi-polynomial time [Marek Cygan et al., 2013].

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Problems, reductions and completeness
• Theory of computation → Approximation algorithms analysis
##### Keywords
• Hardness of Approximation
• Bounded Degree
• Constraint Satisfaction Problems
• Independent Set

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