The best approximation ratio for maximum directed cut

Pin down the best approximation ratio for maximum directed cut, known under the Unique Games Conjecture to lie in [0.87446, 0.87461].

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State the problem, from TCS Open Problems

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	1	+# The best approximation ratio for maximum directed cut
	2	+
	3	+Maximum directed cut asks for a set of vertices that maximizes the total weight of arcs leaving it. Assuming the Unique Games Conjecture, the best ratio a polynomial-time algorithm can guarantee is now known to lie between 0.87446 and 0.87461, and the question is what the exact value is.
	4	+
	5	+## Problem
	6	+
	7	+Let $G=(V,E)$ be a directed graph with nonnegative arc weights $w$. For $S \subseteq V$ let $\delta^+(S)$ be the set of arcs $(u,v)\in E$ with $u \in S$ and $v \notin S$, and let
	8	+
	9	+$$\mathrm{OPT}(G) = \max_{S \subseteq V} w\big(\delta^+(S)\big).$$
	10	+
	11	+A randomized polynomial-time algorithm is an $\alpha$-approximation for MAX DI-CUT if on every instance it outputs a set $S$ with $\mathbb{E}\,[w(\delta^+(S))] \ge \alpha \cdot \mathrm{OPT}(G)$. Equivalently, MAX DI-CUT is the constraint satisfaction problem whose constraints have the form $\bar x_u \wedge x_v$ on Boolean variables. Let $\alpha_{\text{DI-CUT}}$ be the best ratio achievable in polynomial time, assuming the Unique Games Conjecture (UGC).
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	13	+Question. Determine $\alpha_{\text{DI-CUT}}$. It is known that $0.87446 \le \alpha_{\text{DI-CUT}} \le 0.87461$.
	14	+
	15	+## Background
	16	+
	17	+Goemans and Williamson's semidefinite programming method (1995) gives a ratio $\alpha_{\text{GW}} \approx 0.87856$ for undirected MAX CUT, and Khot, Kindler, Mossel and O'Donnell showed that, under UGC, nothing better is possible [4, 5]. Feige and Goemans, who improved the algorithms for the directed version [3], asked whether it can be approximated as well as MAX CUT [1]. Raghavendra's theorem says that under UGC a canonical SDP with the right rounding is optimal for every such problem, but it gives no practical way to compute the resulting number, so the exact ratio for MAX DI-CUT is still unknown. The question was posted on TCS Open Problems in 2026.
	18	+
	19	+## What is known
	20	+
	21	+- Upper bound. Assuming UGC, $\alpha_{\text{DI-CUT}} \le 0.87461$ (Brakensiek, Huang, Potechin and Zwick, 2023) [1]. Since this is below $\alpha_{\text{GW}}$, MAX DI-CUT is strictly harder to approximate than MAX CUT, which answers Feige and Goemans's question. The proof is analytical, and the authors expect more elaborate constructions to lower it slightly.
	22	+- Lower bound. $\alpha_{\text{DI-CUT}} \ge 0.87446$ [1], by a rounding scheme that picks a random threshold function from a distribution found by computer search; the ratio is certified with interval arithmetic, and the code is public ([jbrakensiek/max-dicut](https://github.com/jbrakensiek/max-dicut)). The previous best, about 0.87401, came from a single threshold rounding of Lewin, Livnat and Zwick (2002) [2].
	23	+- A related problem. For MAX 2-AND, where negated variables are allowed, $0.87414 \le \alpha_{\text{2AND}} \le 0.87435$ [1], so MAX DI-CUT is strictly easier than MAX 2-AND, refuting a conjecture of Austrin.
	24	+- Without UGC. Approximating MAX DI-CUT within $12/13+\varepsilon$ is NP-hard (Trevisan, Sorkin, Sudan and Williamson, 2000) [6].
	25	+
	26	+## What would count as progress
	27	+
	28	+- A rounding scheme whose ratio is certified above 0.87446, for example with the interval-arithmetic verification of [1]. The authors conjecture that distributions over threshold roundings are enough to reach the optimum.
	29	+- A distribution of SDP configurations that defeats every rounding scheme, giving a UGC-hardness bound below 0.87461.
	30	+- A proof that determines $\alpha_{\text{DI-CUT}}$ exactly, or a better unconditional NP-hardness bound than $12/13$.
	31	+
	32	+## References
	33	+
	34	+1. J. Brakensiek, N. Huang, A. Potechin and U. Zwick, Separating MAX 2-AND, MAX DI-CUT and MAX CUT, FOCS 2023; SIAM J. Comput. 55(3) (2026). [arXiv:2212.11191](https://arxiv.org/abs/2212.11191)
	35	+2. M. Lewin, D. Livnat and U. Zwick, Improved rounding techniques for the MAX 2-SAT and MAX DI-CUT problems, IPCO 2002, LNCS 2337, 67–82. [doi:10.1007/3-540-47867-1_6](https://doi.org/10.1007/3-540-47867-1_6)
	36	+3. U. Feige and M. Goemans, Approximating the value of two prover proof systems, with applications to MAX 2SAT and MAX DICUT, ISTCS 1995, 182–189. [doi:10.1109/ISTCS.1995.377033](https://doi.org/10.1109/ISTCS.1995.377033)
	37	+4. M. X. Goemans and D. P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM 42(6) (1995), 1115–1145. [doi:10.1145/227683.227684](https://doi.org/10.1145/227683.227684)
	38	+5. S. Khot, G. Kindler, E. Mossel and R. O'Donnell, Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?, SIAM J. Comput. 37(1) (2007), 319–357. [doi:10.1137/S0097539705447372](https://doi.org/10.1137/S0097539705447372)
	39	+6. L. Trevisan, G. B. Sorkin, M. Sudan and D. P. Williamson, Gadgets, approximation, and linear programming, SIAM J. Comput. 29(6) (2000), 2074–2097. [doi:10.1137/S0097539797328847](https://doi.org/10.1137/S0097539797328847)
	40	+
	41	+## Source
	42	+
	43	+Imported from [TCS Open Problems, Problem 1](https://tcsopenproblems.com/problem/1), curated by Kunal Marwaha. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.