A combinatorial algorithm for MAX CSP

• Authors: Mayur Datar;Tomás Feder;Aristides Gionis;Rajeev Motwani;Rina Panigrahy
• Venue: Information Processing Letters
• Year: 2003

We consider the problem MAX CSP over multi-valued domains with variables ranging over sets of size si ≤ s and constraints involving kj ≤ k variables. We study two algorithms with approximation ratios A and B. respectively, so we obtain a solution with approximation ratio max (A, B).The first algorithm is based on the linear programming algorithm of Serna, Trevisan, and Xhafa [Proc. 15th Annual Symp. on Theoret. Aspects of Comput. Sci., 1998, pp. 488-498] and gives ratio A which is bounded below by s1-k. For k = 2, our bound in terms of the individual set sizes is the minimum over all constraints involving two variables of (1/2√s1+ 1/2√s2)2, where s1 and s2 are the set sizes for the two variables.We then give a simple combinatorial algorithm which has approximation ratio B, with B A/e. The bound is greater than s1-k/e in general, and greater than s1-k(1 - (s - 1)/2(k - 1)) for s ≤ k - 1, thus close to the s1-k linear programming bound for large k. For k = 2, the bound is 4/9 if s = 2, 1/2(s - 1) if s ≥ 3, and in general greater than the minimum of 1/4S1 + 1/4s2 over constraints with set sizes s1 and s2, thus within a factor of two of the linear programming bound.For the case of k = 2 and s = 2 we prove an integrality gap of 4/9 (1 + O(n-1/2)). This shows that our analysis is tight for any method that uses the linear programming upper bound.