Optimal Streaming Approximations for all Boolean Max-2CSPs

04/24/2020 ∙ by Chi-Ning Chou, et al. ∙ 0

We prove tight upper and lower bounds on approximation ratios of all Boolean Max-2CSP problems in the streaming model. Specifically, for every type of Max-2CSP problem, we give an explicit constant α, s.t. for any ϵ>0 (i) there is an (α-ϵ)-streaming approximation using space O(logn); and (ii) any (α+ϵ)-streaming approximation requires space Ω(√(n)). This generalizes the celebrated work of [Kapralov, Khanna, Sudan SODA 2015; Kapralov, Krachun STOC 2019], who showed that the optimal approximation ratio for Max-CUT was 1/2. Prior to this work, the problem of determining this ratio was open for all other Max-2CSPs. Our results are quite surprising for some specific Max-2CSPs. For the Max-DCUT problem, there was a gap between an upper bound of 1/2 and a lower bound of 2/5 [Guruswami, Velingker, Velusamy APPROX 2017]. We show that neither of these bounds is tight, and the optimal ratio for Max-DCUT is 4/9. We also establish that the tight approximation for Max-2SAT is √(2)/2, and for Exact Max-2SAT it is 3/4. As a byproduct, our result gives a separation between space-efficient approximations for Max-2SAT and Exact Max-2SAT. This is in sharp contrast to the setting of polynomial-time algorithms with polynomial space, where the two problems are known to be equally hard to approximate.

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