Simultaneous Approximation of Constraint Satisfaction Problems

TL;DR

This paper introduces the first nontrivial approximation algorithms for simultaneous constraint satisfaction problems, achieving constant factor Pareto approximations for small k, and exploring the limits of such approximations.

Contribution

It presents the first polynomial-time approximation algorithms for simultaneous Max-$F$-CSPs for small k, extending to Max-w-SAT, and establishes hardness results for larger k.

Findings

Polynomial-time constant factor Pareto approximation for k < ~log^{1/4} n

Improved approximation for simultaneous Max-w-SAT for k < ~log^{1/3} n

Hardness results for k = ω(log n) assuming ETH

Abstract

Given $k$ collections of 2SAT clauses on the same set of variables $V$, can we find one assignment that satisfies a large fraction of clauses from each collection? We consider such simultaneous constraint satisfaction problems, and design the first nontrivial approximation algorithms in this context. Our main result is that for every CSP $F$, for $k < \tilde{O}(\log^{1/4} n)$, there is a polynomial time constant factor Pareto approximation algorithm for $k$ simultaneous Max-$F$-CSP instances. Our methods are quite general, and we also use them to give an improved approximation factor for simultaneous Max-w-SAT (for $k <\tilde{O}(\log^{1/3} n)$). In contrast, for $k = \omega(\log n)$, no nonzero approximation factor for $k$ simultaneous Max-$F$-CSP instances can be achieved in polynomial time (assuming the Exponential Time Hypothesis). These problems are a natural meeting point for the theory of constraint satisfaction problems and multiobjective optimization. We also suggest a number of interesting directions for future research.