A Boolean maximum constraint satisfaction problem, Max-CSP(f ), is a maximization problem specified by a constraint function f : {−1, 1} k → {0, 1}. An instance of Max-CSP(f ) consists of n variables and m constraints, where each constraint is f applied on a tuple of "literals" of k distinct variables chosen from the n variables. f is said to be symmetric if f (x) depends only on k i=1 x i , where x = (x 1 , . . . , x k ). Chou, Golovnev, and Velusamy [CGV20] obtained explicit constants characterizing the streaming approximability of all symmetric Max-2CSPs. More recently, Chou, Golovnev, Sudan, and Velusamy [CGSV21a] proved a general dichotomy theorem showing tight 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 ǫ > 0, Max-CSP(f ) is (α(f ) − ǫ)-approximable by a linear sketching algorithm in O(log n) space, but any (α(f ) + ǫ)-approximation sketching algorithm for Max-CSP(f ) requires Ω( √ n)space. While they show that α(f ) is computable to arbitrary precision in PSPACE, they do not give a closed-form expression.In this work, we build on the [CGSV21a] dichotomy theorem and give closed-form expressions for the sketching approximation ratios of multiple families of symmetric Boolean functions. These include kAND and Th k−1
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