Yumou Fei

Yumou Fei, Dor Minzer, Shuo Wang

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 $\mathcal{F}$ of $k$-ary predicates over $Σ$ and any $c\in (0,1)$, there exists $0<s\leq c$ such that: 1. For any $\varepsilon>0$ there is a constant pass, $O_{\varepsilon}(\log n)$-space randomized streaming algorithm solving the promise problem $\text{MaxCSP}(\mathcal{F})[c,s-\varepsilon]$. That is, the algorithm accepts inputs with value at least $c$ with probability at least $2/3$, and rejects inputs with value at most $s-\varepsilon$ with probability at least $2/3$. 2. For all $\varepsilon>0$, any $p$-pass (even randomized) streaming algorithm that solves the promise problem $\text{MaxCSP}(\mathcal{F})[c,s+\varepsilon]$ must use $Ω_{\varepsilon}(n^{1/3}/p)$ space. Our approximation algorithm is based on a certain linear-programming relaxation of the CSP and on a distributed algorithm that approximates its value. This part builds on the works [Yoshida, STOC 2011] and [Saxena, Singer, Sudan, Velusamy, SODA 2025]. For our hardness result we show how to translate an integrality gap of the linear program into a family of hard instances, which we then analyze via studying a related communication complexity problem. That analysis is based on discrete Fourier analysis and builds on a prior work of the authors and on the work [Chou, Golovnev, Sudan, Velusamy, J.ACM 2024].

Yumou Fei, Dor Minzer, Shuo Wang

In a streaming constraint satisfaction problem (streaming CSP), a $p$-pass algorithm receives the constraints of an instance sequentially, making $p$ passes over the input in a fixed order, with the goal of approximating the maximum fraction of satisfiable constraints. We show near optimal space lower bounds for streaming CSPs, improving upon prior works. (1) Fei, Minzer and Wang (\textit{STOC 2026}) showed that for any CSP, the basic linear program defines a threshold $α_{\mathrm{LP}}\in [0,1]$ such that, for any $\varepsilon > 0$, an $(α_{\mathrm{LP}} - \varepsilon)$-approximation can be achieved using constant passes and polylogarithmic space, whereas achieving $(α_{\mathrm{LP}} + \varepsilon)$-approximation requires $Ω(n^{1/3}/p)$ space. We improve this lower bound to $Ω(\sqrt{n}/p)$, which is nearly tight for a gap version of the problem. (2) For $p=o(\log n)$, we further strengthen the lower bound to $Ω(n\cdot2^{-O_{\varepsilon}(p)})$. Combined with existing algorithmic results, this shows that $α_{\mathrm{LP}}$ is not only the limit of multi-pass polylogarithmic-space algorithms, but also the limit of single-pass sublinear-space algorithms on bounded-degree instances. (3) For certain CSPs, we show that there exists $α< 1$ such that achieving an $α$-approximation requires $Ω(n/p)$ space. Our proofs are Fourier analytic, building on the techniques of Fei, Minzer and Wang (\textit{STOC 2026}) and the Fourier-$\ell_1$-based lower bound method of Kapralov and Krachun (\textit{STOC 2019}).

Yumou Fei

We initiate the study of distribution testing for probability distributions over the edges of a graph, motivated by the closely related question of ``edge-distribution-free'' graph property testing. The main results of this paper are nearly-tight bounds on testing bipartiteness, triangle-freeness and square-freeness of edge distributions, whose sample complexities are shown to scale as $Θ(n)$, $n^{4/3\pm o(1)}$ and $n^{9/8\pm o(1)}$, respectively. The technical core of our paper lies in the proof of the upper bound for testing square-freeness, wherein we develop new techniques based on certain birthday-paradox-type lemmas that may be of independent interest. We will discuss how our techniques fit into the general framework of distribution-free property testing. We will also discuss how our results are conceptually connected with Turán problems and subgraph removal lemmas in extremal combinatorics.