Document

# Improved NP-Inapproximability for 2-Variable Linear Equations

## File

LIPIcs.APPROX-RANDOM.2015.341.pdf
• Filesize: 0.55 MB
• 20 pages

## Cite As

Johan Håstad, Sangxia Huang, Rajsekar Manokaran, Ryan O’Donnell, and John Wright. Improved NP-Inapproximability for 2-Variable Linear Equations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 341-360, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.341

## Abstract

An instance of the 2-Lin(2) problem is a system of equations of the form "x_i + x_j = b (mod 2)". Given such a system in which it's possible to satisfy all but an epsilon fraction of the equations, we show it is NP-hard to satisfy all but a C*epsilon fraction of the equations, for any C < 11/8 = 1.375 (and any 0 < epsilon <= 1/8). The previous best result, standing for over 15 years, had 5/4 in place of 11/8. Our result provides the best known NP-hardness even for the Unique Games problem, and it also holds for the special case of Max-Cut. The precise factor 11/8 is unlikely to be best possible; we also give a conjecture concerning analysis of Boolean functions which, if true, would yield a larger hardness factor of 3/2. Our proof is by a modified gadget reduction from a pairwise-independent predicate. We also show an inherent limitation to this type of gadget reduction. In particular, any such reduction can never establish a hardness factor C greater than 2.54. Previously, no such limitation on gadget reductions was known.
##### Keywords
• approximability
• unique games
• linear equation
• linear programming

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Amit Agarwal, Moses Charikar, Konstantin Makarychev, and Yury Makarychev. O(√log n) approximation algorithms for Min-Uncut, Min-2CNF-Deletion, and directed cut problems. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pages 573-581, 2005.
2. Sanjeev Arora, Boaz Barak, and David Steurer. Subexponential algorithms for Unique Games and related problems. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, pages 563-572, 2010.
3. Sanjeev Arora, Satish Rao, and Umesh Vazirani. Expander flows, geometric embeddings and graph partitioning. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pages 222-231, 2004.
4. Per Austrin and Johan Håstad. On the usefulness of predicates. ACM Trans. Comput. Theory, 5(1):1:1-1:24, 2013.
5. Mihir Bellare, Oded Goldreich, and Madhu Sudan. Free bits, PCPs, and non-approximability - towards tight results. SIAM Journal of Computing, 27(3):804-915, 1998.
6. Siu On Chan. Approximation resistance from pairwise independent subgroups. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing, pages 447-456, 2013.
7. Moses Charikar, Konstantin Makarychev, and Yury Makarychev. Near-optimal algorithms for Unique Games. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pages 205-214, 2006.
8. Miroslav Chlebík and Janka Chlebíková. On approximation hardness of the minimum 2SAT-DELETION problem. In Proceedings of the 29th Annual International Symposium on Mathematical Foundations of Computer Science, pages 263-273, 2004.
9. Irit Dinur and Samuel Safra. On the hardness of approximating minimum vertex cover. Annals of Mathematics, 162(1):439-485, 2005.
10. Uriel Feige and Daniel Reichman. On systems of linear equations with two variables per equation. In Proceedings of the 7th Annual International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, pages 117-127, 2004.
11. Michel Goemans and David Williamson. A 0.878 approximation algorithm for MAX-2SAT and MAX-CUT. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 422-431, 1994.
12. Anupam Gupta, Kunal Talwar, and David Witmer. Sparsest Cut on bounded treewidth graphs: algorithms and hardness results. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing, pages 281-290, 2013.
13. Johan Håstad. Some optimal inapproximability results. Journal of the ACM, 48(4):798-859, 2001.
14. Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pages 767-775, 2002.
15. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for Max-Cut and other 2-variable CSPs? SIAM Journal on Computing, 37(1):319-357, 2007.
16. Dana Moshkovitz and Ran Raz. Two-query PCP with subconstant error. Journal of the ACM, 57(5):29, 2010.
17. Elchanan Mossel, Ryan O'Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions with low influences: invariance and optimality. Annals of Mathematics, 171(1):295-341, 2010.
18. Ryan O'Donnell and John Wright. A new point of NP-hardness for Unique-Games. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing, pages 289-306, 2012.
19. Anup Rao. Parallel repetition in projection games and a concentration bound. SIAM Journal of Computing, 40(6):1871-1891, 2011.
20. Luca Trevisan, Gregory Sorkin, Madhu Sudan, and David Williamson. Gadgets, approximation, and linear programming. SIAM Journal on Computing, 29(6):2074-2097, 2000.
21. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theoretical Computer Science, 348(2-3):357-365, 2005.