Max-DICUT: Directed Cut Optimization

• Max-DICUT is a directed cut problem that maximizes the normalized count of edges crossing a bipartition in a directed graph, serving as a canonical case for approximation studies.
• It employs advanced algorithmic methods like SDP, LP, and oblivious rounding to achieve near-optimal approximations despite its NP-hard complexity.
• The problem is pivotal in streaming algorithm research, where sublinear-space and multi-pass strategies are developed to approximate solutions under adversarial inputs.

The Max-DICUT (Maximum Directed Cut) problem is a central optimization problem in theoretical computer science and combinatorial optimization, with deep connections to constraint satisfaction, streaming models, and approximation algorithms. Its study is motivated by both its intrinsic theoretical appeal and its role as a canonical test case for advanced algorithmic paradigms, particularly within the context of streaming computation and approximation resistance of Boolean @@@@2@@@@.

1. Formal Definition and Basic Properties

Let $G = (V, E)$ be a directed graph with $n = |V|$ vertices and $m = |E|$ directed edges (arcs), allowing for multi-edges but not self-loops. An ordered bipartition $(L, R)$ of $V$ defines a directed cut (dicut); an edge $u \to v \in E$ belongs to the dicut if $u \in L$ and $v \in R$. The normalized cut value is: $$\mathrm{DICUT}(G; L, R) = \frac{|\{u \to v \in E : u \in L, v \in R\}|}{m}$$ The maximum directed cut value is: $$\mathrm{Max}(G) = \max_{(L, R)} \mathrm{DICUT}(G; L, R)$$ which is the objective in both non-weighted and weighted formulations (with analogous normalization by total edge-weight in the latter). The Max-DICUT problem is to find a cut achieving $\mathrm{Max}(G)$, or, more typically, to efficiently compute a multiplicative or additive approximation thereof.

2. Approximability and Complexity

The Max-DICUT problem is NP-hard. In undirected graphs, the Max-Cut problem admits a 0.878-approximation by the Goemans-Williamson SDP rounding scheme, a ratio that is tight under UGC (Nakajima et al., 12 Feb 2024). For Max-DICUT, the best known polynomial-time approximation algorithm attains a ratio in the interval $0.87446 \leq \alpha_{\mathrm{DI}} \leq 0.87461$ under UGC, strictly below the Max-Cut constant (Brakensiek et al., 2022). This separation was established using both a randomized rounding scheme based on mixtures of threshold functions, and a UGC-hardness construction employing carefully designed SDP configurations.

The table below summarizes the key approximation ratios under UGC:

Problem	Worst-case Approx. Ratio	Reference
Max-Cut	≈ 0.87856	[Goemans-Williamson, KKMO07]
Max-DICUT	0.87446 – 0.87461	(Brakensiek et al., 2022)
Max-2-AND	< 0.87435	[Austrin, (Brakensiek et al., 2022)]

The integrality gap of the standard LP relaxation for Max-DICUT is exactly 2, even for $\Omega(n^c)$ rounds of Sherali-Adams (Kenkre et al., 2015).

3. Algorithmic Methods: SDP, LP, and Oblivious Rounding

Semidefinite Programming: The best approximation algorithm for Max-DICUT solves a Goemans–Williamson-style SDP with weighted “triangle” constraints. The unique directionality of arcs is encoded using constraints on the vectors’ dot-products, and optimal rounding is achieved via carefully designed randomized threshold-based procedures, surpassing earlier LP-based or naïve randomized methods (Nakajima et al., 12 Feb 2024, Brakensiek et al., 2022).

Oblivious Algorithms: Feige and Jozeph introduced oblivious algorithms, which independently assign vertices to one side of the cut using a function of their local bias, defined as

$b(v) = \frac{\deg^+(v)}{\deg^+(v) + \deg^-(v)}$

where $\deg^+$ and $\deg^-$ are out-degree and in-degree, respectively. The class of oblivious algorithms is characterized by a selection function $f:[0,1] \to [0,1]$, with $p_v = f(b(v))$ and enforced antisymmetry $f(x) + f(1-x) = 1$ (Feige et al., 2010). The optimal oblivious algorithm achieves ratio ≈ 0.4836, while the trivial (uninformed) random cut achieves 1/4, and no oblivious algorithm can beat 1/2 due to even cycle instances.

4. Streaming and Sublinear-Space Algorithms

Max-DICUT is a canonical instance for the study of streaming algorithms, especially in sublinear space regimes. The streaming model allows the input edge set to be provided as a stream, possibly in adversarial or random order, seeking to output a multiplicative (or additive) approximation to the optimum in one or more passes and space $o(n)$.

Space/Pass Complexity Landscape

• Single-Pass: A sharp lower bound of 1/2 is known; no $o(n)$-space single-pass algorithm can outperform this approximation, even for bounded degree, by [Kapralov–Krachun STOC’19, (Velusamy, 22 Dec 2025)].
• Two-Pass: Saxena, Singer, Sudan, Velusamy [SODA’25] provide a (1/2–ε)-approximation in sublinear space for bounded-degree in a single pass; for arbitrary (unbounded-degree) graphs, O(1/ε) passes are required (Velusamy, 22 Dec 2025). The main advance is a two-pass, $O(n^{1-\delta})$-space algorithm for arbitrary graphs that achieves (1/2–ε)-approximation (additive ε error) (Velusamy, 22 Dec 2025).
• Oblivious Streaming: Classic oblivious algorithms can be implemented in streaming via sketching—partition the bias range, bucketize vertices, and estimate cut densities between buckets. O(log n) space suffices for a 4/9 approximation in adversarial order; $\widetilde{O}(\sqrt{n})$-space enables 0.483-approximation via higher-order “smoothed snapshot” estimation (Saxena et al., 2022).
• Random-Order vs Adversarial-Order: In the random-order model, O(log n)-space single-pass algorithms achieve the optimal (for oblivious) 0.483-approximation (Saxena et al., 2022). This is strictly better than the 4/9 barrier for o(√n)-space in adversarial order.

Advanced Streaming Algorithmic Ingredients

The leading two-pass algorithm (Velusamy, 22 Dec 2025) combines:

• Trevisan’s bounded-degree reduction (liftings to bounded-degree graphs with preserved Max-DICUT value).
• SSSV’s single-pass bounded-degree simulation of LOCAL algorithms via neighborhood sampling and type reweighting.
• Bhaskara–Dharuki–Venkatasubramanian’s core-set algorithm for the dense regime.

5. Streaming Algorithmic Schematics and Complexity

For arbitrary directed graphs on n vertices:

• Passes: 2
• Approximation: For every constant ε > 0, achieves (1/2–ε)-approximation up to additive ε, i.e.,

$$\frac{1}{2} \mathrm{Max}(G) - \epsilon \leq v \leq \mathrm{Max}(G) + \epsilon$$

with probability at least 2/3 (Velusamy, 22 Dec 2025).

• Space: O(n^{1–δ}), where δ = δ(ε) > 0 depends only on ε.
• Key Steps:
  • Pass 1: Sample copies of vertices and build approximate degree sketches.
  • Pass 2: For sparse regime, reconstruct a random bounded-degree lift via local sampling, and simulate distributed approximations via neighborhood types; for dense regime, construct core-sets and solve offline on compact summaries.

The algorithms crucially make use of probabilistic concentration (Chernoff/Hoeffding bounds) to guarantee correctness of the computations and approximation of the requisite polytope/graph structure (Velusamy, 22 Dec 2025).

6. Relation to Other CSPs and Separations

Max-DICUT is a prototypical example of a Boolean 2-CSP with significant implications for CSP approximability hierarchies. In terms of UGC-based hardness, it is strictly easier than Max-2-AND and strictly harder than Max-Cut (Brakensiek et al., 2022). The recent precise separation: $$\alpha_{2\mathrm{AND}} < \alpha_{\mathrm{DI}} < \alpha_{\mathrm{CUT}}$$ is achieved via explicit analytical threshold rounding and computer-assisted worst-case instance analysis. Furthermore, extensions to Max-k-AND elucidate that for $k$-ary CSPs, gap amplification and streaming complexity results generalize key aspects of the Max-DICUT case, with “factor-revealing” linear programs capturing the worst-case performance of oblivious algorithms (Singer, 2023).

7. Impact, Limitations, and Open Directions

The study of Max-DICUT anchors several major themes in modern theoretical computer science:

• Streaming Lower Bounds: The 1/2 single-pass lower bound is tight for o(n)-space algorithms; the two-pass (1/2–ε) result is optimal within this marginal increase in passes (Velusamy, 22 Dec 2025).
• Algorithmic Techniques: Smoothed snapshot estimation, degree reduction, and simulating global SDP or LP rounding locally in constrained models are all advanced tools with impact beyond Max-DICUT (Saxena et al., 2022, Velusamy, 22 Dec 2025).
• Approximation Resistance: Max-DICUT fails to reach the approximation threshold of Max-Cut, with current best algorithms matching UGC-based hardness up to O(10^{-4}) (Brakensiek et al., 2022). The optimal margin of improvement, if any, likely requires further structural insights or entirely new algorithmic paradigms.
• Promise CSPs: Under certain “promise” conditions, such as the presence of a large directed cut, deterministic polynomial-time algorithms can output undirected cuts of value at least as large as the directional optimum—yielding tight “ratio = 1” for this restricted setting (Nakajima et al., 12 Feb 2024).

Open questions persist regarding optimal one-pass approximability in various streaming models (e.g., for restricted classes of graphs), structural gadget reductions for new CSP hierarchies, and the extension of snapshot and sketching methods to higher-arity or non-Boolean CSPs. The Max-DICUT problem continues to serve as a focal point for advancing both algorithmic and complexity-theoretic frontiers.