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

• Authors: Subhash Khot;Guy Kindler;Elchanan Mossel;Ryan O'Donnell
• Venue: FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
• Year: 2004

In this paper we give evidence suggesting that MAXCUT is NP-hard to approximate to within a factor of 驴_GW + 驴, for all 驴 0, where 驴_GW denotes the approximation ratio achieved by the Goemans-Williamson algorithm [14], 驴_GW 驴 .878567. This result is conditional, relying on two conjectures: a) the Unique Games conjecture of Khot [24]; 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 β + e)-approximate MAX-2SAT, where β 驴 .943943 is the minimum of({{2 + \frac{2}{\pi }\theta )} \mathord{\left/ {\vphantom {{2 + \frac{2}{\pi }\theta )} {(3 - \cos (\theta ))}}} \right. \kern-\nulldelimiterspace} {(3 - \cos (\theta ))}} on (\frac{\pi }{2},\pi ). Motivated by our proof techniques, we show that if the MAX-2CSP and MAX-2SAT problems are slightly restricted 驴 in a way that seems to retain all their hardness 驴 then they have (驴_GW - 驴)- and (β - 驴)-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 (\frac{3}{4} + \frac{1}{{2\pi }} + \varepsilon )-approximate (驴 .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.