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We explore the use of local algorithms in the design of streaming algorithms for the Maximum Directed Cut problem. Specifically, building on the local algorithm of (Buchbinder, Feldman, Seffi, and Schwartz [14] and Censor-Hillel, Levy, and Shachnai [16]), we develop streaming algorithms for both adversarially and randomly ordered streams that approximate the value of maximum directed cut in bounded-degree graphs. In n-vertex graphs, for adversarially ordered streams, our algorithm uses O (n1-Ω(1)) (sub-linear) space and for randomly ordered streams, our algorithm uses logarithmic space. Moreover, both algorithms require only one pass over the input stream. With a constant number of passes, we give a logarithmic-space algorithm which works even on graphs with unbounded degree on adversarially ordered streams. Our algorithms achieve any fixed constant approximation factor less than 1/2. In the single-pass setting, this is tight: known lower bounds show that obtaining any constant approximation factor greater than 1/2 is impossible without using linear space in adversarially ordered streams (Kapralov and Krachun [37]) and space in randomly ordered streams, even on bounded degree graphs (Kapralov, Khanna, and Sudan [35]). In terms of techniques, our algorithms partition the vertices into a small number of different types based on the structure of their local neighborhood, ensuring that each type carries enough information about the structure to approximately simulate the local algorithm on a vertex with that type. We then develop tools to accurately estimate the frequency of each type. This allows us to simulate an execution of the local algorithm on all vertices, and thereby approximate the value of the maximum directed cut. 

We explore the use of local algorithms in the design of streaming algorithms for the Maximum Directed Cut problem. Specifically, building on the local algorithm of (Buchbinder, Feldman, Seffi, and Schwartz [14] and Censor-Hillel, Levy, and Shachnai [16]), we develop streaming algorithms for both adversarially and randomly ordered streams that approximate the value of maximum directed cut in bounded-degree graphs. In n-vertex graphs, for adversarially ordered streams, our algorithm uses O (n1-Ω(1)) (sub-linear) space and for randomly ordered streams, our algorithm uses logarithmic space. Moreover, both algorithms require only one pass over the input stream. With a constant number of passes, we give a logarithmic-space algorithm which works even on graphs with unbounded degree on adversarially ordered streams. Our algorithms achieve any fixed constant approximation factor less than 1/2. In the single-pass setting, this is tight: known lower bounds show that obtaining any constant approximation factor greater than 1/2 is impossible without using linear space in adversarially ordered streams (Kapralov and Krachun [37]) and space in randomly ordered streams, even on bounded degree graphs (Kapralov, Khanna, and Sudan [35]). In terms of techniques, our algorithms partition the vertices into a small number of different types based on the structure of their local neighborhood, ensuring that each type carries enough information about the structure to approximately simulate the local algorithm on a vertex with that type. We then develop tools to accurately estimate the frequency of each type. This allows us to simulate an execution of the local algorithm on all vertices, and thereby approximate the value of the maximum directed cut. 

Abstract An ordering constraint satisfaction problem (OCSP) is defined by a family$$\mathcal F$$ $F$of predicates mapping permutations on$$\{1,\ldots,k\}$$ $\{1,\dots ,k\}$to$$\{0,1\}$$ $\{0,1\}$. An instance of ($$\mathcal F$$ $F$) onnvariables consists of a list of constraints, each consisting of a predicate from$$\mathcal F$$ $F$applied onkdistinct variables. The goal is to find an ordering of thenvariables that maximizes the number of constraints for which the induced ordering on thekvariables satisfies the predicate. OCSPs capture well-studied problems including ‘maximum acyclic subgraph’ () and “maximum betweenness”. In this work, we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, when an instance is presented as a stream of constraints. We show that for every$$\mathcal F$$ $F$, ($$\mathcal F$$ $F$) is approximation-resistant to o(n)-space streaming algorithms, i.e., algorithms using o(n) space cannot distinguish streams where almost every constraint is satisfiable from streams where no ordering beats the random ordering by a noticeable amount. This space bound is tight up to polylogarithmic factors. In the case of , our result shows that for every$$\epsilon>0$$ $ϵ>0$, is not$$(1/2+\epsilon)$$ $(1/2+ϵ)$-approximable in o(n) space. The previous best inapproximability result, due to Guruswami & Tao (2019), only ruled out 3/4-approximations in$$o(\sqrt n)$$ $o\left(\sqrt{n}\right)$space. Our results build on recent works of Chou et al. (2022b, 2024) who provide a tight, linear-space inapproximability theorem for a broad class of “standard” (i.e., non-ordering) constraint satisfaction problems (CSPs) over arbitrary (finite) alphabets. Our results are obtained by building a family of appropriate standard CSPs (one for every alphabet sizeq) from any given OCSP and applying their theorem to this family of CSPs. To convert the resulting hardness results for standard CSPs back to our OCSP, we show that the hard instances from this earlier theorem have the following “partition expansion” property with high probability: For every partition of thenvariables into small blocks, for most of the constraints, all variables are in distinct blocks. 

We give an $$\widetilde{O}(\sqrt{n})$$-space single-pass 0.483-approximation streaming algorithm for estimating the maximum directed cut size (Max-DICUT) in a directed graph on n vertices. This improves over an $$O(\log n)$$-space $4 / 9 < 0.45$ approximation algorithm due to Chou, Golovnev, and Velusamy (FOCS 2020), which was known to be optimal for $$o(\sqrt{n})$$-space algorithms. Max-DICUT is a special case of a constraint satisfaction problem (CSP). In this broader context, we give the first CSP for which algorithms with $$\widetilde{O}(\sqrt{n})$$- space can provably outperform $$o(\sqrt{n})$$- space algorithms. The key technical contribution of our work is development of the notions of a first-order snapshot of a (directed) graph and of estimates of such snapshots. These snapshots can be used to simulate certain (non-streaming) Max-DICUT algorithms, including the “oblivious” algorithms introduced by Feige and Jozeph (Algorithmica, 2015), who showed that one such algorithm Previous work of the authors (SODA 2023) studied the restricted case of bounded-degree graphs, and observed that in this setting, it is straightforward to estimate the snapshot with $$\ell_{1}$$ errors and this suffices to simulate oblivious algorithms. But for unbounded-degree graphs, even defining an achievable and sufficient notion of estimation is subtle. We describe a new notion of snapshot estimation and prove its sufficiency using careful smoothing techniques, and then develop an algorithm which sketches such an estimate via a delicate process of intertwined vertex- and edge-subsampling. Prior to our work, the only streaming algorithms for any CSP on general instances were based on generalizations of the $$O(\log n)$$-space algorithm for Max-DICUT, and can roughly be characterized as based on “zeroth” order snapshots. Our work thus opens the possibility of a new class of algorithms for approximating CSPs by demonstrating that more sophisticated snapshots can outperform cruder ones in the case of Max-DICUT.