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1. Streaming complexity of CSPs with randomly ordered constraints

   Saxena, Raghuvansh R.; Singer, Noah; Sudan, Madhu; Velusamy, Santhoshini (January 2023, Proceedings of the 2023 {ACM-SIAM} Symposium on Discrete Algorithms, {SODA} 2023)

   Nikhil, Bansal; Nagarajan, Viswanath (Ed.)

   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. 

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2. On Sketching Approximations for Symmetric Boolean CSPs

   Boyland, Joanna; Hwang, Michael; Prasad, Tarun; Singer, Noah; Velusamy, Santhoshini (September 2022, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022))

   Chakrabarti, Amit; Swamy, Chaitanya (Ed.)

   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. 

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3. Linear space streaming lower bounds for approximating CSPs

   https://doi.org/10.1145/3519935.3519983  

   Chou, Chi-Ning; Golovnev, Alexander; Sudan, Madhu; Velingker, Ameya; Velusamy, Santhoshini (June 2022, {STOC} '22: 54th Annual {ACM} {SIGACT} Symposium on Theory of Computing)

   Leonardi, Stefano; Gupta, Anupam (Ed.)

   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. 

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4. Streaming Algorithms via Local Algorithms for Maximum Directed Cut

   Saxena, Raghuvansh R; Singer, Noah G; Sudan, Madhu; Velusamy, Santhoshini (January 2025, Society for Industrial and Applied Mathematics)

   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. 

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5. Streaming Algorithms via Local Algorithms for Maximum Directed Cut

   Saxena, Raghuvansh R; Singer, Noah G; Sudan, Madhu; Velusamy, Santhoshini (January 2025, Society for Industrial and Applied Mathematics)

   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. 

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