On Some Recent MAX SAT Approximation Algorithms

TL;DR

This paper reviews recent randomized 3/4-approximation algorithms for MAX SAT, simplifies their analysis, shows their equivalence, and extends them to deterministic LP rounding methods.

Contribution

It simplifies the analysis of existing MAX SAT approximation algorithms, proves their equivalence, and extends them to deterministic LP rounding techniques.

Findings

Algorithms are all structurally similar and achieve 3/4 approximation.

The Buchbinder et al. algorithm is equivalent to Van Zuylen's.

Extension to deterministic LP rounding algorithms is demonstrated.

Abstract

Recently a number of randomized 3/4-approximation algorithms for MAX SAT have been proposed that all work in the same way: given a fixed ordering of the variables, the algorithm makes a random assignment to each variable in sequence, in which the probability of assigning each variable true or false depends on the current set of satisfied (or unsatisfied) clauses. To our knowledge, the first such algorithm was proposed by Poloczek and Schnitger; Van Zuylen subsequently gave an algorithm that set the probabilities differently and had a simpler analysis. She also set up a framework for deriving such algorithms. Buchbinder, Feldman, Naor, and Schwartz, as a special case of their work on maximizing submodular functions, also give a randomized 3/4-approximation algorithm for MAX SAT with the same structure as these previous algorithms. In this note we give a gloss on the Buchbinder et al. algorithm that makes it even simpler, and show that in fact it is equivalent to the previous algorithm of Van Zuylen. We also show how it extends to a deterministic LP rounding algorithm; such an algorithm was also given by Van Zuylen.