Optimal Inapproximability Results for Max-Cut and Other 2-Variable CSPs?

Abstract

In this paper, we give evidence suggesting that MAX-CUT is NP-hard to approximate to within a factor of /spl alpha//sub cw/+ /spl epsi/, for all /spl epsi/ > 0, where /spl alpha//sub cw/ denotes the approximation ratio achieved by the Goemans-Williamson algorithm (1995). /spl alpha//sub cw/ /spl ap/ .878567. This result is conditional, relying on two conjectures: a) the unique games conjecture of Khot; and, b) a very believable conjecture we call the majority is stablest conjecture. These results indicate that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-CUT problem. The same two conjectures also imply that it is NP-hard to (/spl beta/ + /spl epsi/)-approximate MAX-2SAT, where /spl beta/ /spl ap/ .943943 is the minimum of (2 + (2//spl pi/) /spl theta/)/(3 - cos(/spl theta/)) on (/spl pi//2, /spl pi/). Motivated by our proof techniques, we show that if the MAX-2CSP and MAX-2SAT problems are slightly restricted - in a way that seems to retain all their hardness -then they have (/spl alpha//sub GW/-/spl epsi/)- and (/spl beta/ - /spl epsi/)-approximation algorithms, respectively. Though we are unable to prove the majority is stablest conjecture, we give some partial results and indicate possible directions of attack. Our partial results are enough to imply that MAX-CUT is hard to (3/4 + 1/(2/spl pi/) + /spl epsi/)-approximate (/spl ap/ .909155), assuming only the unique games conjecture. We also discuss MAX-2CSP problems over non-Boolean domains and state some related results and conjectures. We show, for example, that the unique games conjecture implies that it is hard to approximate MAX-2LIN(q) to within any constant factor.