# 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*k*CSP. We also show a connection between probabilistic 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.

## Keywords

Feasible Solution Approximate Algorithm Semidefinite Programming Linear Programming Relaxation Probabilistic Proof Check## Preview

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