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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. 

We initiate a study of the streaming complexity of constraint satisfaction problems (CSPs) when the constraints arrive in a random order. We show that there exists a CSP, namely Max-DICUT, for which random ordering makes a provable difference. Whereas a 4/9 ≈ 0.445 approximation of DICUT requires space with adversarial ordering, we show that with random ordering of constraints there exists a 0.483-approximation algorithm that only needs O(log n) space. We also give new algorithms for Max-DICUT in variants of the adversarial ordering setting. Specifically, we give a two-pass O(log n) space 0.483-approximation algorithm for general graphs and a single-pass space 0.483-approximation algorithm for bounded-degree graphs. On the negative side, we prove that CSPs where the satisfying assignments of the constraints support a one-wise independent distribution require -space for any non-trivial approximation, even when the constraints are randomly ordered. This was previously known only for adversarially ordered constraints. Extending the results to randomly ordered constraints requires switching the hard instances from a union of random matchings to simple Erdős-Renyi random (hyper)graphs and extending tools that can perform Fourier analysis on such instances. The only CSP to have been considered previously with random ordering is Max-CUT where the ordering is not known to change the approximability. Specifically it is known to be as hard to approximate with random ordering as with adversarial ordering, for space algorithms. Our results show a richer variety of possibilities and motivate further study of CSPs with randomly ordered constraints. 

A Boolean maximum constraint satisfaction problem, Max-CSP(f), is specified by a predicate f:{-1,1}^k → {0,1}. An n-variable instance of Max-CSP(f) consists of a list of constraints, each of which applies f to k distinct literals drawn from the n variables. For k = 2, Chou, Golovnev, and Velusamy [Chou et al., 2020] obtained explicit ratios characterizing the √ n-space streaming approximability of every predicate. For k ≥ 3, Chou, Golovnev, Sudan, and Velusamy [Chou et al., 2022] proved a general dichotomy theorem for √ n-space sketching algorithms: For every f, there exists α(f) ∈ (0,1] such that for every ε > 0, Max-CSP(f) is (α(f)-ε)-approximable by an O(log n)-space linear sketching algorithm, but (α(f)+ε)-approximation sketching algorithms require Ω(√n) space. In this work, we give closed-form expressions for the sketching approximation ratios of multiple families of symmetric Boolean functions. Letting α'_k = 2^{-(k-1)} (1-k^{-2})^{(k-1)/2}, we show that for odd k ≥ 3, α(kAND) = α'_k, and for even k ≥ 2, α(kAND) = 2α'_{k+1}. Thus, for every k, kAND can be (2-o(1))2^{-k}-approximated by O(log n)-space sketching algorithms; we contrast this with a lower bound of Chou, Golovnev, Sudan, Velingker, and Velusamy [Chou et al., 2022] implying that streaming (2+ε)2^{-k}-approximations require Ω(n) space! We also resolve the ratio for the "at-least-(k-1)-1’s" function for all even k; the "exactly-(k+1)/2-1’s" function for odd k ∈ {3,…,51}; and fifteen other functions. We stress here that for general f, the dichotomy theorem in [Chou et al., 2022] only implies that α(f) can be computed to arbitrary precision in PSPACE, and thus closed-form expressions need not have existed a priori. Our analyses involve identifying and exploiting structural "saddle-point" properties of this dichotomy. Separately, for all threshold functions, we give optimal "bias-based" approximation algorithms generalizing [Chou et al., 2020] while simplifying [Chou et al., 2022]. Finally, we investigate the √ n-space streaming lower bounds in [Chou et al., 2022], and show that they are incomplete for 3AND, i.e., they fail to rule out (α(3AND})-ε)-approximations in o(√ n) space. 

We analyze the sketching approximability of constraint satisfaction problems on Boolean domains, where the constraints are balanced linear threshold functions applied to literals. In particular, we explore the approximability of monarchy-like functions where the value of the function is determined by a weighted combination of the vote of the first variable (the president) and the sum of the votes of all remaining variables. The pure version of this function is when the president can only be overruled by when all remaining variables agree. For every k ≥ 5, we show that CSPs where the underlying predicate is a pure monarchy function on k variables have no non-trivial sketching approximation algorithm in o(√n) space. We also show infinitely many weaker monarchy functions for which CSPs using such constraints are non-trivially approximable by O(log(n)) space sketching algorithms. Moreover, we give the first example of sketching approximable asymmetric Boolean CSPs. Our results work within the framework of Chou, Golovnev, Sudan, and Velusamy (FOCS 2021) that characterizes the sketching approximability of all CSPs. Their framework can be applied naturally to get a computer-aided analysis of the approximability of any specific constraint satisfaction problem. The novelty of our work is in using their work to get an analysis that applies to infinitely many problems simultaneously. 

We consider the approximability of constraint satisfaction problems in the streaming setting. For every constraint satisfaction problem (CSP) on n variables taking values in {0,…,q−1}, we prove that improving over the trivial approximability by a factor of q requires Ω(n) space even on instances with O(n) constraints. We also identify a broad subclass of problems for which any improvement over the trivial approximability requires Ω(n) space. The key technical core is an optimal, q−(k−1)-inapproximability for the Max k-LIN-mod q problem, which is the Max CSP problem where every constraint is given by a system of k−1 linear equations mod q over k variables. Our work builds on and extends the breakthrough work of Kapralov and Krachun (Proc. STOC 2019) who showed a linear lower bound on any non-trivial approximation of the MaxCut problem in graphs. MaxCut corresponds roughly to the case of Max k-LIN-mod q with k=q=2. For general CSPs in the streaming setting, prior results only yielded Ω(√n) space bounds. In particular no linear space lower bound was known for an approximation factor less than 1/2 for any CSP. Extending the work of Kapralov and Krachun to Max k-LIN-mod q to k>2 and q>2 (while getting optimal hardness results) is the main technical contribution of this work. Each one of these extensions provides non-trivial technical challenges that we overcome in this work.