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

arXiv2212.11191MaRDI QIDQ6421270

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Publication date: 21 December 2022

Abstract: Assuming the Unique Games Conjecture (UGC), the best approximation ratio that can be obtained in polynomial time for the MAX CUT problem is $alph{a}_{extCUT}simeq0.87856$, 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 $0.87446lealph{a}_{extDI-CUT}le0.87461$, where $alph{a}_{extDI-CUT}$ 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 $z_{1}land{z}_2$, where $z_1$ and $z_2$ are literals, i.e., variables or their negations (In MAX DI-CUT each constraint is of the form $$, where $x_1$ and $x_2$ are variables.) Austrin separated MAX 2-AND from MAX CUT by showing that $alph{a}_{ext2AND}<0.87435$ 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 $0.87414lealph{a}_{ext2AND}le0.87435$. 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.

Has companion code repository: https://github.com/jbrakensiek/max-dicut

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