In a constraint satisfaction problem (CSP) in the single-pass streaming model, an algorithm is given the constraints C1,…,Cm of an instance one after another (in some fixed order), and its goal is to approximate the value of the instance, i.e., the maximum fraction of constraints that can be satisfied simultaneously. In the p-pass streaming model the algorithm is given p passes over the input stream (in the same order), after which it is required to output an approximation of the value of the instance. We show a dichotomy result for p-pass streaming algorithms for all CSPs and for up to polynomially many passes. More precisely, we prove that for any arity parameter k, finite alphabet Σ, collection F of k-ary predicates over Σ and any c∈ (0,1), there exists 0

We study the spectral gap of subgraphs of the hypercube induced by monotone subsets of vertices. For a monotone subset $A\subseteq\{0,1\}^{n}$ of density $μ(A)$, the previous best lower bound on the spectral gap, due to Cohen, was $γ\gtrsim μ(A)/n^{2}$, improving upon the earlier bound $γ\gtrsim μ(A)^{2}/n^{2}$ established by Ding and Mossel. In this paper, we prove the optimal lower bound $γ\gtrsim μ(A)/n$. As a corollary, we improve the mixing time upper bound of the random walk on constant-density monotone sets from $O(n^{3})$, as shown by Ding and Mossel, to $O(n^{2})$. Along the way, we develop two new inequalities that may be of independent interest: (1)~a directed $L^{2}$-Poincaré inequality on the hypercube, and (2)~an ``approximate'' FKG inequality for monotone sets.

In the Max-Cut problem in the streaming model, an algorithm is given the edges of an unknown graph $G=(V, E)$ in some fixed order, and its goal is to approximate the size of the largest cut in G. Improving upon an earlier result of Kapralov, Khanna and Sudan, it was shown by Kapralov and Krachun that for all $\varepsilon\gt 0$, no $o(n)$ memory streaming algorithm can achieve a $(1 / 2+\varepsilon)$-approximation for Max-Cut. Their result holds for single-pass streams, i.e. the setting in which the algorithm only views the stream once, and it was open whether multi-pass access may help. The state-of-the-art result along these lines, due to Assadi and N, rules out arbitrarily good approximation algorithms with constantly many passes and $n^{1-\delta}$ space for any $\delta\gt 0$. We improve upon this state-of-the-art result, showing that any non-trivial approximation algorithm for Max-Cut requires either polynomially many passes or polynomially large space. More specifically, we show that for all $\varepsilon\gt 0$, a k-pass streaming $(1 / 2+\varepsilon)$-approximation algorithm for Max-Cut requires $\Omega_{\varepsilon}\left(n^{1 / 3} / k\right)$ space. This result leads to a similar lower bound for the Maximum Directed Cut problem, showing the near optimality of the algorithm of [Saxena, Singer, Sudan, Velusamy, SODA 2025]. Our lower bounds proceed by showing a communication complexity lower bound for the Distributional Implicit Hidden Partition (DIHP) Problem, introduced by Kapralov and Krachun. While a naive application of the discrepancy method fails, we identify a property of protocols called “globalness”, and show that (1) any protocol for DIHP can be turned into a global protocol, (2) the discrepancy of a global protocol must be small. The second step is the more technically involved step in the argument, and therein we use global hypercontractive inequalities, and more specifically strong quantitative versions of the level- d inequality for global functions.

We study the approximability of computing the partition functions of two-state spin systems. The problem is parameterized by a 2 × 2 symmetric matrix. Previous results on this problem were restricted either to the case where the matrix has non-negative entries, or to the case where the diagonal entries are equal, i.e. Ising models. In this paper, we study the generalization to arbitrary 2 × 2 interaction matrices with real entries. We show that in some regions of the parameter space, it's #P-hard to even determine the sign of the partition function, while in other regions there are fully polynomial approximation schemes for the partition function. Our results reveal several new computational phase transitions.

We study the approximability of computing the partition functions of two-state spin systems. The problem is parameterized by a 2×2 symmetric matrix. Previous results on this problem were restricted either to the case where the matrix has non-negative entries, or to the case where the diagonal entries are equal, i.e. Ising models. In this paper, we study the generalization to arbitrary 2×2 interaction matrices with real entries. We show that in some regions of the parameter space, it’s #P-hard to even determine the sign of the partition function, while in other regions there are fully polynomial approximation schemes for the partition function. Our results reveal several new computational phase transitions.

We give a distribution-free testing algorithm for decision lists with Õ(n11/12/ε3) queries. This is the first sublinear algorithm for this problem, which shows that, unlike halfspaces, testing is strictly easier than learning for decision lists. Complementing the algorithm, we show that any distribution-free tester for decision lists must make Ω(√n) queries, or draw Ω(n) samples when the algorithm is sample-based.