Positive Linear Programming, Parallel Approximation and PCP's

Several sequential approximation algorithms are based on the following paradigm: solve a linear or semideenite 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 nding some new linear programming relaxations, we develop improved parallel approximation algorithms for Max Sat, Max DiCut, and Max k-CSP. We also show a connection between prob-abilistic proof checking and a restricted version of Max k-CSP. This implies that our approximation algorithm for Max k-CSP can be used to prove inclusion in P for certain PCP classes.