Closed-form expressions for the sketching approximability of (some) symmetric Boolean CSPs
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A Boolean maximum constraint satisfaction problem, Max-CSP(f), is specified by a constraint function f:{-1,1}^k→{0,1}; an instance on n variables is given by a list of constraints applying f on a tuple of "literals" of k distinct variables chosen from the n variables. Chou, Golovnev, and Velusamy [CGV20] obtained explicit constants characterizing the streaming approximability of all symmetric Max-2CSPs. More recently, Chou, Golovnev, Sudan, and Velusamy [CGSV21] proved a general dichotomy theorem tightly characterizing the approximability of Boolean Max-CSPs with respect to sketching algorithms. For every f, they showed that there exists an optimal approximation ratio α(f)∈ (0,1] such that for every ϵ>0, Max-CSP(f) is (α(f)-ϵ)-approximable by a linear sketching algorithm in O(log n) space, but any (α(f)+ϵ)-approximation sketching algorithm for Max-CSP(f) requires Ω(√(n)) space. In this work, we build on the [CGSV21] dichotomy theorem and give closed-form expressions for the sketching approximation ratios of multiple families of symmetric Boolean functions. The functions include kAND and Th_k^k-1 (the “weight-at-least-(k-1)” threshold function on k variables). In particular, letting α'_k = 2^-(k-1) (1-k^-2)^(k-1)/2, we show that for odd k ≥ 3, α(kAND= α'_k; for even k ≥ 2, α(kAND) = 2α'_k+1; and for even k ≥ 2, α(Th_k^k-1) = k/2α'_k-1. We also resolve the ratio for the “weight-exactly-k+1/2” function for odd k ∈{3,…,51} as well as fifteen other functions. These closed-form expressions need not have existed just given the [CGSV21] dichotomy. For arbitrary threshold functions, we also give optimal "bias-based" approximation algorithms generalizing [CGV20] and simplifying [CGSV21].
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