Separating MAX 2-AND, MAX DI-CUT and MAX CUT

Abstract

Assuming the Unique Games Conjecture (UGC), the best approximation ratio that can be obtained in polynomial time for the MAX CUT problem is ${\alpha }_{\text{CUT}}\simeq 0.87856$, obtained by the celebrated SDP-based approximation algorithm of Goemans and Williamson. The currently best approximation algorithm for MAX DI-CUT, i.e., the MAX CUT problem in directed graphs, achieves a ratio of about $0.87401$, leaving open the question whether MAX DI-CUT can be approximated as well as MAX CUT. We obtain a slightly improved algorithm for MAX DI-CUT and a new UGC-hardness result for it, showing that $0.87446\le {\alpha }_{\text{DI-CUT}}\le 0.87461$, where ${\alpha }_{\text{DI-CUT}}$ is the best approximation ratio that can be obtained in polynomial time for MAX DI-CUT under UGC. The new upper bound separates MAX DI-CUT from MAX CUT, resolving a question raised by Feige and Goemans. A natural generalization of MAX DI-CUT is the MAX 2-AND problem in which each constraint is of the form $z_1\wedge z_2$, where $z_1$ and $z_2$ are literals, i.e., variables or their negations (In MAX DI-CUT each constraint is of the form ${\bar{x}}_1\wedge x_2$, where $x_1$ and $x_2$ are variables.) Austrin separated MAX 2-AND from MAX CUT by showing that ${\alpha }_{\text{2AND}}<0.87435$ and conjectured that MAX 2-AND and MAX DI-CUT have the same approximation ratio. Our new lower bound on MAX DI-CUT refutes this conjecture, completing the separation of the three problems MAX 2-AND, MAX DI-CUT and MAX CUT. We also obtain a new lower bound for MAX 2-AND, showing that $0.87414\le {\alpha }_{\text{2AND}}\le 0.87435$. Our upper bound on MAX DI-CUT is achieved via a simple, analytical proof. The lower bounds on MAX DI-CUT and MAX 2-AND (the new approximation algorithms) use experimentally-discovered distributions of rounding functions which are then verified via computer-assisted proofs.

39 pages, 5 figures, 7 tables