Sketching approximations and LP approximations for finite CSPs are related

Noah G. Singer; Madhur Tulsiani; Santhoshini Velusamy

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arXiv · 2509.17926

Sketching approximations and LP approximations for finite CSPs are related

Noah G. Singer·Madhur Tulsiani·Santhoshini Velusamy

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Abstract

We identify a connection between the approximability of CSPs in two models: (i) sublinear space streaming algorithms, and (ii) the basic LP relaxation. We show that whenever the basic LP admits an integrality gap, there is an $\Omega(\sqrt{n})$-space sketching lower bound. We also show that all existing linear space streaming lower bounds for Max-CSPs can be lifted to integrality gap instances for basic LPs. For bounded-degree graphs, by combining the distributed algorithm of Yoshida (STOC 2011) for approximately solving the basic LP with techniques described in Saxena, Singer, Sudan, and Velusamy (SODA 2025) for simulating a distributed algorithm by a sublinear space streaming algorithm on bounded-degree instances of Max-DICUT, it appears that there are sublinear space streaming algorithms implementing the basic LP, for every CSP. Based on our results, we conjecture the following dichotomy theorem: Whenever the basic LP admits an integrality gap, there is a linear space single-pass streaming lower bound, and when the LP is roundable, there is a sublinear space streaming algorithm.

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Noah G. Singer, Madhur Tulsiani, Santhoshini Velusamy. 2025-09-22. Sketching approximations and LP approximations for finite CSPs are related. https://arxiv.org/abs/2509.17926

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