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On the NP-Hardness Approximation Curve for Max-2Lin(2)

Author Björn Martinsson

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Björn Martinsson
  • KTH Royal Institute of Technology, Stockholm, Sweden

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Björn Martinsson. On the NP-Hardness Approximation Curve for Max-2Lin(2). In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 11:1-11:38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.11

Abstract

In the Max-2Lin(2) problem you are given a system of equations on the form x_i + x_j ≡ b mod 2, and your objective is to find an assignment that satisfies as many equations as possible. Let c ∈ [0.5, 1] denote the maximum fraction of satisfiable equations. In this paper we construct a curve s (c) such that it is NP-hard to find a solution satisfying at least a fraction s of equations. This curve either matches or improves all of the previously known inapproximability NP-hardness results for Max-2Lin(2). In particular, we show that if c ⩾ 0.9232 then (1 - s(c))/(1 - c) > 1.48969, which improves the NP-hardness inapproximability constant for the min deletion version of Max-2Lin(2). Our work complements the work of O'Donnell and Wu that studied the same question assuming the Unique Games Conjecture. Similar to earlier inapproximability results for Max-2Lin(2), we use a gadget reduction from the (2^k - 1)-ary Hadamard predicate. Previous works used k ranging from 2 to 4. Our main result is a procedure for taking a gadget for some fixed k, and use it as a building block to construct better and better gadgets as k tends to infinity. Our method can be used to boost the result of both smaller gadgets created by hand (k = 3) or larger gadgets constructed using a computer (k = 4).

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Constraint and logic programming
  • Theory of computation → Graph algorithms analysis
Keywords
  • Inapproximability
  • NP-hardness
  • 2Lin(2)
  • Max-Cut
  • Gadget

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References

  1. David Applegate, William Cook, Sanjeeb Dash, and Daniel Espinoza. Qsopt_ex rational lp solver. https://www.math.uwaterloo.ca/~bico/qsopt/ex/. Accessed: 2023-09-20.
  2. Sanjeev Arora, Boaz Barak, and David Steurer. Subexponential algorithms for unique games and related problems. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, volume 62, pages 563-572, October 2010. URL: https://doi.org/10.1109/FOCS.2010.59.
  3. Per Austrin and Johan Håstad. On the usefulness of predicates. In 2012 IEEE 27th Conference on Computational Complexity, pages 53-63, 2012. URL: https://doi.org/10.1109/CCC.2012.18.
  4. Per Austrin and Elchanan Mossel. Approximation resistant predicates from pairwise independence. CoRR, abs/0802.2300, 2008. URL: https://arxiv.org/abs/0802.2300.
  5. Mihir Bellare, Oded Goldreich, and Madhu Sudan. Free bits, pcps, and nonapproximability - towards tight results. SIAM Journal on Computing, 27(3):804-915, 1998. URL: https://doi.org/10.1137/S0097539796302531.
  6. Siu On Chan. Approximation resistance from pairwise independent subgroups. In Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, STOC '13, pages 447-456, New York, NY, USA, 2013. Association for Computing Machinery. URL: https://doi.org/10.1145/2488608.2488665.
  7. Uriel Feige and Michael Langberg. The rpr2 rounding technique for semidefinite programs. Journal of Algorithms, 60(1):1-23, 2006. URL: https://doi.org/10.1016/j.jalgor.2004.11.003.
  8. Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42(6):1115-1145, November 1995. URL: https://doi.org/10.1145/227683.227684.
  9. Johan Håstad. Some optimal inapproximability results. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, STOC '97, pages 1-10, New York, NY, USA, 1997. Association for Computing Machinery. URL: https://doi.org/10.1145/258533.258536.
  10. Johan Håstad, Sangxia Huang, Rajsekar Manokaran, Ryan O’Donnell, and John Wright. Improved NP-Inapproximability for 2-Variable Linear Equations. In Naveen Garg, Klaus Jansen, Anup Rao, and José D. P. Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015), volume 40 of Leibniz International Proceedings in Informatics (LIPIcs), pages 341-360, Dagstuhl, Germany, 2015. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.341.
  11. S. Khot, G. Kindler, E. Mossel, and R. O'Donnell. Optimal inapproximability results for max-cut and other 2-variable csps? In 45th Annual IEEE Symposium on Foundations of Computer Science, pages 146-154, 2004. URL: https://doi.org/10.1109/FOCS.2004.49.
  12. Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, STOC '02, pages 767-775, New York, NY, USA, 2002. Association for Computing Machinery. URL: https://doi.org/10.1145/509907.510017.
  13. Subhash Khot, Dor Minzer, and Muli Safra. Pseudorandom sets in grassmann graph have near-perfect expansion. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 592-601, October 2018. URL: https://doi.org/10.1109/FOCS.2018.00062.
  14. Ryan O'Donnell and Yi Wu. An optimal sdp algorithm for max-cut, and equally optimal long code tests. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, STOC '08, pages 335-344, New York, NY, USA, 2008. Association for Computing Machinery. URL: https://doi.org/10.1145/1374376.1374425.
  15. L. Trevisan, G.B. Sorkin, M. Sudan, and D.P. Williamson. Gadgets, approximation, and linear programming. In Proceedings of 37th Conference on Foundations of Computer Science, pages 617-626, 1996. URL: https://doi.org/10.1109/SFCS.1996.548521.
  16. Mårten Wiman. Improved inapproximability of max-cut through min-cut. Master’s thesis, KTH, School of Electrical Engineering and Computer Science (EECS), 2018. Google Scholar
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