Approximation of non-boolean 2CSP

We develop a polynomial time Ω ( 1 R logR ) approximate algorithm for Max 2CSP-R, the problem where we are given a collection of constraints, each involving two variables, where each variable ranges over a set of size R, and we want to find an assignment to the variables that maximizes the number of satisfied constraints. Assuming the Unique Games Conjecture, this is the best possible approximation up to constant factors. Previously, a 1/R-approximate algorithm was known, based on linear programming. Our algorithm is based on semidefinite programming (SDP) and on a novel rounding technique. The SDP that we use has an almost-matching integrality gap. [email protected]. School of Computer Science and Engineering, Hebrew University. Part of this work was done while visiting the Simons Institute. Supported by Israeli Science Fund Grant No. 1692/13, and Binational Science Foundation Grant No. 2012220. † [email protected]. Computer Science Department, UIUC. Part of this work was done while visiting the Simons Institute. This material is based upon work supported by the National Science Foundation under Grant No. 1423452. ‡ [email protected]. EECS Department and Simons Institute, U.C. Berkeley. This material is based upon work supported by the National Science Foundation under Grant No. 1216642 and by the US-Israel Binational Science Foundation under Grant No. 2010451.