Parallel Approximation Algorithms by Positive Linear Programming

Abstract

Several sequential approximation algorithms for combinatorial optimization problems 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 [LN] 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 Maxk CSP. The Max Sat algorithm essentially matches the best approximation obtainable with sequential algorithms and has a fast sequential version. The Maxk CSP algorithm improves even over previous sequential algorithms. We also show a connection between probabilistic proof checking and a restricted version of Maxk CSP. This implies that our approximation algorithm for Maxk CSP can be used to prove inclusion in P for certain PCP classes.