Positive linear programming, parallel approximation and PCP’s

Abstract

Several sequential approximation algorithms are based on the following paradigm: solve a linear or semidefinite programming relaxation, then use randomized rounding to convert fractional solutions of the relaxation into integer solutions for the original combinatorial problem. We demonstrate that such a paradigm can also yield parallel approximation algorithms by showing how to convert certain linear programming relaxations into essentially equivalent positive linear programming [18] relaxations that can be near-optimally solved in NC. Building on this technique, and finding some new linear programming relaxations, we develop improved parallel approximation algorithms for MAX SAT, MAX DIRECTED CUT, and MAX kCSP. We also show a connection between probabilistic proof checking and a restricted version of MAX kCSP. This implies that our approximation algorithm for Max kCSP can be used to prove inclusion in P for certain PCP classes.