Separating MAX 2-AND, MAX DI-CUT and MAX CUT
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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≃ 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≤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, 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 x̅_1 x_2, where x_1 and x_2 are variables.) Austrin separated MAX 2-AND from MAX CUT by showing that α_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≤α_2AND≤ 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.
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