Abstract
For a prime p, max-2linp is the problem of satisfying as many equations as possible from a system of linear equations modulo p, where every equation contains two variables. Hastad shows that this problem is NP-hard to approximate within a ratio of 11/12 + ε for p=2, and Andersson, Engebretsen and Hastad show the same hardness of approximation ratio for p ≥ 11, and somewhat weaker results (such as 69/70) for p = 3,5,7. We prove that max-2linp is easiest to approximate when p = 2, implying for every prime p that max-2linp is NP-hard to approximate within a ratio of 11/12 + ε. For large p, we prove stronger hardness of approximation results. Namely, we show that there is some universal constant δ > 0 such that it is NP-hard to approximate max-2linp within a ratio better than 1/p δ. We use our results so as to clarify some aspects of Khot’s unique games conjecture. Namely, we show that for every ε > 0 it is NP-hard to approximate the value of unique games within a ratio of ε.
This research was supported by the Israel Science Foundation (Grant number 263/02).
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Feige, U., Reichman, D. (2004). On Systems of Linear Equations with Two Variables per Equation. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 2004 2004. Lecture Notes in Computer Science, vol 3122. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27821-4_11
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DOI: https://doi.org/10.1007/978-3-540-27821-4_11
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