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 α_ CUT ∼eq 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 UG-Chardness result for it, showing that 0.87446 ≤ α_ DI-CUT ≤ 0.87461, where α_ 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 ^ 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 x1 ^ x_2, where x_1 and x_2 are variables.) Austrin separated MAX 2-AND from MAX CUT by showing that α_2 AND ≤ 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 ≤ α_2 AND ≤ 0.87435. Our upper bound on MAXDI-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.¹¹Code for the project: https://github.com/jbrakensiek/max-dicut