Abstract
We study the approximability of the MAX k-CSP problem over non-boolean domains, more specifically over {0,1,...,q − 1} for some integer q. We extend the techniques of Samorodnitsky and Trevisan [19] to obtain a UGC hardness result when q is a prime. More precisely, assuming the Unique Games Conjecture, we show that it is NP-hard to approximate the problem to a ratio greater than q 2 k/q k. Independent of this work, Austrin and Mossel [2] obtain a more general UGC hardness result using entirely different techniques.
We also obtain an approximation algorithm that achieves a ratio of C(q) ·k/q k for some constant C(q) depending only on q, via a subroutine for approximating the value of a semidefinite quadratic form when the variables take values on the corners of the q-dimensional simplex. This generalizes an algorithm of Nesterov [16] for the ±1-valued variables. It has been pointed out to us [15] that a similar approximation ratio can be obtained by reducing the non-boolean case to a boolean CSP.
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Keywords
- Approximation Algorithm
- Constraint Satisfaction
- Constraint Satisfaction Problem
- Hardness Result
- Inapproximability Result
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Guruswami, V., Raghavendra, P. (2008). Constraint Satisfaction over a Non-Boolean Domain: Approximation Algorithms and Unique-Games Hardness. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2008 2008. Lecture Notes in Computer Science, vol 5171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85363-3_7
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DOI: https://doi.org/10.1007/978-3-540-85363-3_7
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