The Approximability of Constraint Satisfaction Problems

Thumbnail Image
Penn collection
Departmental Papers (CIS)
Degree type
approximation algorithms
approximation classes
approximation-preserving reductions
complete problems
Boolean constraint satisfaction problems
hardness of approximation
Grant number
Copyright date
Related resources
Sudan, Madhu
Trevisan, Luca
Williamson, David P

We study optimization problems that may be expressed as "Boolean constraint satisfaction problems". An instance of a Boolean constraint satisfaction problem is given by m constraints applied to n Boolean variables. Different computational problems arise from constraint satisfaction problems depending on the nature of the "underlying" constraints as well as on the goal of the optimization task. Here we consider four possible goals: MAX CSP (MIN CSP) is the class of problems where the goal is to find an assignment maximizing the number of satisfied constraints (minimizing the number of unsatisfied constraints). MAX ONES (MIN ONES) is the class of optimization problems where the goal is to find an assignment satisfying all constraints with maximum (minimum) number of variables set to 1. Each class consists of infinitely many problems and a problem within a class is specified by a finite collection of finite Boolean functions that describe the possible constraints that may be used. Tight bounds on the approximability of every problem in MAX CSP were obtained by Creignou [11]. In this work we determine tight bounds on the "approximability" (i.e., the ratio to within which each problem may be approximated in polynomial time) of every problem in MAX ONES, MIN CSP and MIN ONES. Combined with the result of Creignou, this completely classifies all optimization problems derived from Boolean constraint satisfaction. Our results capture a diverse collection of optimization problems such as MAX 3-SAT, MAX CUT, MAX CLIQUE, MIN CUT, NEAREST CODEWORD etc. Our results unify recent results on the (in)approximability of these optimization problems and yield a compact presentation of most known results. Moreover, these results provide a formal basis to many statements on the behavior of natural optimization problems, that have so far only been observed empirically.

Date Range for Data Collection (Start Date)
Date Range for Data Collection (End Date)
Digital Object Identifier
Series name and number
Publication date
Journal title
Volume number
Issue number
Publisher DOI
Journal Issue
Copyright SIAM, 2000. Postprint version. Published in SIAM Journal on Computing, Volume 30, Number 6, January 2000, pages 1863-1920. Publisher URL:
Postprint version. Copyright ACM, 2000. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in SIAM Journal on Computing, Volume 30, Number 6, January 2000, pages 1863-1920. Publisher URL:
Recommended citation