Constraint Satisfaction over a Non-Boolean Domain: Approximation Algorithms and Unique-Games Hardness

  • Authors:

  • Venkatesan Guruswami;Prasad Raghavendra

  • Affiliations:

  • Department of Computer Science & Engineering, University of Washington, Seattle;Department of Computer Science & Engineering, University of Washington, Seattle

  • Venue:
  • APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
  • Year:
  • 2008

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Abstract

We study the approximability of the MAX k-CSPproblem over non-boolean domains, more specifically over {0,1,...,q茂戮驴 1} for some integer q. We extend the techniques of Samorodnitsky and Trevisan [19] to obtain a UGC hardness result when qis 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 q2k/qk. 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/qkfor 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.