A Dichotomy Theorem for Multi-Pass Streaming CSPs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912997154226176 |
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| author | Fei, Yumou Minzer, Dor Wang, Shuo |
| author_facet | Fei, Yumou Minzer, Dor Wang, Shuo |
| contents | 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]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_11399 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Dichotomy Theorem for Multi-Pass Streaming CSPs Fei, Yumou Minzer, Dor Wang, Shuo Computational Complexity Data Structures and Algorithms 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]. |
| title | A Dichotomy Theorem for Multi-Pass Streaming CSPs |
| topic | Computational Complexity Data Structures and Algorithms |
| url | https://arxiv.org/abs/2509.11399 |