A complete classification of the approximability of maximization problems derived from Boolean constraint satisfaction

In this paper we study the approximability of boolean constraint satisfaction problems. A problem in this class consists of some collection of “constraints” (i.e., functions ~ : {O, I}k ~ {0, l}); an instance of a problem is a set of constraints applied to specified subsets of n boolean variables. Schaefer earlier studied the question of whether one could find in polynomial time a setting of the variables satisfying all constraints; he showed that every such problem is either in P or is NP-complete. We consider optimization variants of these problems in which one either tries to maximize the number of satisfied constraints (as in MAX 3SAT or MAX CUT) or tries to find an assignment satisfying all constraints which maximizes the number of variables set to 1 (aa in MAX CUT or MAX CLIQUE). We completely classify the approximability of all such problems. In the first case, we show that any such optimization problem is either in P or is MAX SNP-hard. In the second case, we show that such problems fall precisely into one of five classes, assuming P # NP: solvable in polynomialtime, approximable to within constant factors in polynomial time (but no better), approximable to within polynomial factors in polynomial time (but no better), not approximable to within any factor but decidable in polynomial time, and not decidable in polynomial time. This result proves formally for this class of problems two results which to this point have only been empirical observations; namely, that NP-hard problems in “sanjeev@research bell-labs. corn. Fundamental Mathematics Research Department, Bell Labs, 700 Mountain Avenue, NJ 07974. This work was performed when the author was at the Department of Computer Science, Stanford University,Stanford, CA 94305. He was supported by a SchlumbergerFoundation Fellowship, an OTL grant, and NSF Grant CCR-9357849. tmadhu~rratson. itm. corn. IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598. $dpw@watson. ibm.corn. IBM Thomas J. Watson ResearchCenter, P.O. Box 218, Yorktown Heights, NY 10!VJR PcrrnwsmlIi)I]mLcdigll;tl 11A cx)pids01’:111(N’Ixlrl 1)1’1111s n):llcri:lt Lw pdwml or clwsr<x)lll usc is yr:mlcd lvilhoul h pn)fidcd 11,:11 Ihc copies arc not mdc or dislrihlll.xl Iil I’ protil or ctmmkv’ci:ll dwtllngc Ihc copyright nolicc. lllc Iillc ol’lllc Illlldic:llitm nnd its Lhlc :Ippcar. am{ nolicc is givm 11):11 copyright is hy pcrmissl(m t)l’lhc .\CA1. Inc. ‘1’0ct)py olherwisc. W repuhlisI1. 10 p<xl <mswvtrs llr 10 rcdislrihufc 10 lists. rcqllircs spwilic pwmissi<m mdhlrlie N()( “ 97 El 1’,1s0,“1’c,;lsL!s,\ Copyrighl I 997 z\C\l ()-S’)7()I-XSX-(1,0705 ..$3.50 MAX SNP always turn out to be MAX SNP-hard, and that there seem to be no natural maximization problems approximable to within polylogarithmic factors but no better.