Approximability of all Boolean CSPs with linear sketches

Chou, Chi-Ning; Golovnev, Alexander; Sudan, Madhu; Velusamy, Santhoshini

doi:10.48550/arxiv.2102.12351

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“…Finally, we investigate the streaming lower bounds in [CGSV21a]. We show that for Th 3 4 , the [CGSV21a] streaming and sketching lower bounds match (previously only known for 2AND), while for 3AND, their streaming lower bound is provably weaker.…”

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confidence: 78%

“…In two recent works [CGSV21a; CGSV21b], Chou, Golovnev, Sudan, and Velusamy proved socalled dichotomy theorems for sketching CSPs. [CGSV21a] dealt with CSPs over Boolean alphabets with negations, while [CGSV21b] dealt with the more general case of CSPs over finite alphabets. 1 [CGSV21a] is most relevant for our purposes, as it concerns Boolean CSPs.…”

Section: Prior Work and Motivationsmentioning

confidence: 99%

“…In this problem, the algorithm's input instance Ψ is promised to either have value ≤ β or value ≥ γ, and the goal is to decide which is the case with probability at least 2 3 . By standard arguments, the minimum approximation ratio α(f ) for Max-CSP(f ) equals the infimum of β γ over all (γ, β) such that Max-CSP(f ) is (γ, β)-approximable (see [CGSV21a,Proposition 2.10] for details).…”

Section: Introductionmentioning

confidence: 99%

“…For arbitrary threshold functions, we also give optimal "biasbased" approximation algorithms generalizing [CGV20] while simplifying [CGSV21a]. Finally, we investigate the streaming lower bounds in [CGSV21a]. We show that for Th 3 4 , the [CGSV21a] streaming and sketching lower bounds match (previously only known for 2AND), while for 3AND, their streaming lower bound is provably weaker.…”

mentioning

confidence: 96%

“…We stress here that the closed-form expressions need not have existed just given the [CGSV21a] dichotomy, and our analysis involves identifying structural "saddle-point" properties of the dichotomy. For arbitrary threshold functions, we also give optimal "biasbased" approximation algorithms generalizing [CGV20] while simplifying [CGSV21a]. Finally, we investigate the streaming lower bounds in [CGSV21a].…”

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confidence: 99%

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On sketching approximations for symmetric Boolean CSPs

Boyland¹,

Hwang²,

Tarun³

et al. 2021

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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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confidence: 78%

Section: Prior Work and Motivationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 96%

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confidence: 99%

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