Assuming the Unique Games Conjecture (UGC), the best approximation ratio that can be obtained in polynomial time for the MAX CUT problem is αCUT ≃ 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 ≤ αDI-CUT ≤ 0.87461, where αDI-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 z1∧z2, where z1 and z2 are literals, i.e., variables or their negations (In MAX DI-CUT each constraint is of the form \neg{x1}∧x2, where x1 and x2 are variables.) Austrin separated MAX 2-AND from MAX CUT by showing that α2AND < 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 ≤ α2AND ≤ 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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An Experimental Evaluation of Semidefinite Programming and Spectral Algorithms for Max Cut
We experimentally evaluate the performance of several Max Cut approximation algorithms. In particular, we compare the results of the Goemans and Williamson algorithm using semidefinite programming with Trevisan’s algorithm using spectral partitioning. The former algorithm has a known .878 approximation guarantee whereas the latter has a .614 approximation guarantee. We investigate whether this gap in approximation guarantees is evident in practice or whether the spectral algorithm performs as well as the SDP. We also compare the performances to the standard greedy Max Cut algorithm which has a .5 approximation guarantee and two additional spectral algorithms. The algorithms are tested on Erdős-Renyi random graphs, complete graphs from TSPLIB, and real-world graphs from the Network Repository. We find, unsurprisingly, that the spectral algorithms provide a significant speed advantage over the SDP. In our experiments, the spectral algorithms return cuts with values which are competitive with those of the SDP.
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- Award ID(s):
- 2007009
- NSF-PAR ID:
- 10340106
- Editor(s):
- Schulz, Christian; Ucar, Bora
- Date Published:
- Journal Name:
- Leibniz international proceedings in informatics
- Volume:
- 233
- ISSN:
- 1868-8969
- Page Range / eLocation ID:
- 19:1-19:14
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/LIPIcs.SEA.2022.19
