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


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)


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
  • Hardness of Approximation
  • Bounded Degree
  • Constraint Satisfaction Problems
  • Independent Set


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  1. Per Austrin, Subhash Khot, and Muli Safra. Inapproximability of vertex cover and independent set in bounded degree graphs. Theory Comput., 7(1):27-43, 2011. URL: https://doi.org/10.4086/toc.2011.v007a003.
  2. Nikhil Bansal, Anupam Gupta, and Guru Guruganesh. On the Lovász theta function for independent sets in sparse graphs. SIAM J. Comput., 47(3):1039-1055, 2018. URL: https://doi.org/10.1137/15M1051002.
  3. Boaz Barak, Ankur Moitra, Ryan O'Donnell, Prasad Raghavendra, Oded Regev, David Steurer, Luca Trevisan, Aravindan Vijayaraghavan, David Witmer, and John Wright. Beating the random assignment on constraint satisfaction problems of bounded degree. In APPROX, pages 110-123, 2015. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.110.
  4. Piotr Berman. A d/2 approximation for maximum weight independent set in d-claw free graphs. Nord. J. Comput., 7(3):178-184, 2000. Google Scholar
  5. Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, and Luca Trevisan. From gap-exponential time hypothesis to fixed parameter tractable inapproximability: Clique, dominating set, and more. SIAM J. Comput., 49(4):772-810, 2020. URL: https://doi.org/10.1137/18M1166869.
  6. Parinya Chalermsook, Ameet Gadekar, Kamyar Khodamoradi, and Joachim Spoerhase. Independent set in k-claw-free graphs: Conditional χ-boundedness and the power of LP/SDP relaxations, 2023. URL: https://arxiv.org/abs/2308.16033.
  7. Siu On Chan. Approximation resistance from pairwise-independent subgroups. J. ACM, 63(3):27:1-27:32, 2016. URL: https://doi.org/10.1145/2873054.
  8. Maria Chudnovsky and Paul Seymour. Claw-free graphs VI. colouring. Journal of Combinatorial Theory, Series B, 100(6):560-572, 2010. URL: https://doi.org/10.1016/j.jctb.2010.04.005.
  9. Marek Cygan. Improved approximation for 3-dimensional matching via bounded pathwidth local search. In FOCS, pages 509-518, 2013. URL: https://doi.org/10.1109/FOCS.2013.61.
  10. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  11. Marek Cygan, Fabrizio Grandoni, and Monaldo Mastrolilli. How to sell hyperedges: The hypermatching assignment problem. In SODA, pages 342-351, 2013. URL: https://doi.org/10.1137/1.9781611973105.25.
  12. Michael Dinitz, Guy Kortsarz, and Ran Raz. Label cover instances with large girth and the hardness of approximating basic k-spanner. ACM Trans. Algorithms, 12(2):25:1-25:16, 2016. URL: https://doi.org/10.1145/2818375.
  13. Irit Dinur. Mildly exponential reduction from gap 3SAT to polynomial-gap label-cover. Electron. Colloquium Comput. Complex., TR16-128, 2016. URL: https://arxiv.org/abs/TR16-128.
  14. Irit Dinur and Pasin Manurangsi. ETH-hardness of approximating 2-CSPs and directed steiner network. In ITCS, pages 36:1-36:20, 2018. URL: https://doi.org/10.4230/LIPIcs.ITCS.2018.36.
  15. Pavel Dvorák, Andreas Emil Feldmann, Ashutosh Rai, and Pawel Rzazewski. Parameterized inapproximability of independent set in H-free graphs. Algorithmica, 85(4):902-928, 2023. URL: https://doi.org/10.1007/s00453-022-01052-5.
  16. Uriel Feige, Shafi Goldwasser, László Lovász, Shmuel Safra, and Mario Szegedy. Interactive proofs and the hardness of approximating cliques. J. ACM, 43(2):268-292, 1996. URL: https://doi.org/10.1145/226643.226652.
  17. Joel Friedman. A proof of Alon’s second eigenvalue conjecture and related problems. American Mathematical Soc., 2008. Google Scholar
  18. Oded Goldreich and Madhu Sudan. Locally testable codes and pcps of almost-linear length. J. ACM, 53(4):558-655, 2006. URL: https://doi.org/10.1145/1162349.1162351.
  19. Martin Grötschel, László Lovász, and Alexander Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1:169-197, 1981. Google Scholar
  20. Johan Håstad. Clique is hard to approximate within n^1-ε. In FOCS, pages 627-636, 1996. URL: https://doi.org/10.1109/SFCS.1996.548522.
  21. Johan Håstad. On bounded occurrence constraint satisfaction. Inf. Process. Lett., 74(1-2):1-6, 2000. URL: https://doi.org/10.1016/S0020-0190(00)00032-6.
  22. Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, 2001. URL: https://doi.org/10.1145/502090.502098.
  23. Elad Hazan, Shmuel Safra, and Oded Schwartz. On the complexity of approximating k-set packing. Comput. Complex., 15(1):20-39, 2006. URL: https://doi.org/10.1007/s00037-006-0205-6.
  24. Subhash Khot. On the power of unique 2-prover 1-round games. In CCC, page 25, 2002. URL: https://doi.org/10.1109/CCC.2002.1004334.
  25. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM J. Comput., 37(1):319-357, 2007. URL: https://doi.org/10.1137/S0097539705447372.
  26. Guy Kindler, Alexandra Kolla, and Luca Trevisan. Approximation of non-boolean 2CSP. In SODA, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch117.
  27. Bundit Laekhanukit. Parameters of two-prover-one-round game and the hardness of connectivity problems. In SODA, pages 1626-1643, 2014. URL: https://doi.org/10.1137/1.9781611973402.118.
  28. Pasin Manurangsi. A note on degree vs gap of min-rep label cover and improved inapproximability for connectivity problems. Inf. Process. Lett., 145:24-29, 2019. URL: https://doi.org/10.1016/j.ipl.2018.08.007.
  29. Pasin Manurangsi and Prasad Raghavendra. A birthday repetition theorem and complexity of approximating dense CSPs. In ICALP, pages 78:1-78:15, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.78.
  30. Pasin Manurangsi, Aviad Rubinstein, and Tselil Schramm. The Strongish Planted Clique Hypothesis and its consequences. In ITCS, pages 10:1-10:21, 2021. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.10.
  31. Elchanan Mossel. Gaussian bounds for noise correlation of functions. Geometric and Functional Analysis, 19(6):1713-1756, 2010. Google Scholar
  32. Meike Neuwohner. An improved approximation algorithm for the maximum weight independent set problem in d-claw free graphs. In STACS, pages 53:1-53:20, 2021. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.53.
  33. Meike Neuwohner. Passing the limits of pure local search for weighted k-set packing. In SODA, pages 1090-1137, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch41.
  34. Prasad Raghavendra. Optimal algorithms and inapproximability results for every CSP? In STOC, pages 245-254, 2008. URL: https://doi.org/10.1145/1374376.1374414.
  35. Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer, 2003. Google Scholar
  36. Maxim Sviridenko and Justin Ward. Large neighborhood local search for the maximum set packing problem. In ICALP, pages 792-803, 2013. URL: https://doi.org/10.1007/978-3-642-39206-1_67.
  37. Theophile Thiery and Justin Ward. An improved approximation for maximum weighted k-set packing. In SODA, pages 1138-1162, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch42.
  38. Luca Trevisan. Non-approximability results for optimization problems on bounded degree instances. In STOC, pages 453-461, 2001. URL: https://doi.org/10.1145/380752.380839.
  39. David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput., 3(1):103-128, 2007. URL: https://doi.org/10.4086/toc.2007.v003a006.
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