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

Assuming the Unique Games Conjecture (UGC), the best approximation ratio that can be obtained in polynomial time for the MAX CUT problem is alpha(CUT) similar or equal to 0.87856, obtained by the celebrated SDP-based approximation algorithm of Goemans and Williamson. Currently, the 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 <= alpha(DI-CUT) <= 0.87461, where alpha(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, i.e., shows that MAX DI-CUT cannot be approximated as well as 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) boolean AND 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 x(1) boolean AND x(2), where x(1) and x(2) are variables.) Austrin separated MAX 2-AND from MAX CUT by showing that alpha(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 <= alpha(2AND) <= 0.87435. Our upper bound on MAX DI-CUT is achieved via a simple analytical proof. The new lower bounds on MAX DI-CUT and MAX 2-AND, i.e., the new approximation algorithms, use experimentally-discovered distributions of rounding functions which are then verified via computer-assisted proofs.(1)