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

Authors Euiwoong Lee, Pasin Manurangsi

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Author Details

Euiwoong Lee
  • University of Michigan, Ann Arbor, MI, USA
Pasin Manurangsi
  • Google Research, Bangkok, Thailand

Funding

  • Lee, Euiwoong: Supported in part by NSF grant CCF-2236669 and Google.

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.

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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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