Simultaneous Approximation of Constraint Satisfaction Problems

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 ${\mathcal {F}}$, for , there is a polynomial time constant factor Pareto approximation algorithm for k simultaneous Max-${\mathcal {F}}$-CSP instances. Our methods are quite general, and we also use them to give an improved approximation factor for simultaneous Max-w-SAT (for ). In contrast, for $k = \omega (\log n)$, no nonzero approximation factor for k simultaneous Max-${\mathcal {F}}$-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.

Amey Bhangale—Research supported in part by NSF grant CCF-1253886.

Swastik Kopparty—Research supported in part by a Sloan Fellowship and NSF grant CCF-1253886.

Sushant Sachdeva—Research supported by the NSF grants CCF-0832797, CCF-1117309, and Daniel Spielman’s & Sanjeev Arora’s Simons Investigator Grants. Part of this work was done when this author was at the Simons Institute for the Theory of Computing, UC Berkeley, and at the Department of Computer Science, Princeton University.

The rights of this work are transferred to the extent transferable according to title 17 §105 U.S.C.