Revisiting maximum satisfiability and related problems in data streams

Abstract

We revisit the maximum satisfiability problem (Max-SAT) in the data stream model. In this problem, the stream consists of m clauses that are disjunctions of literals drawn from n Boolean variables. The objective is to find an assignment to the variables that maximizes the number of satisfied clauses. Chou et al. (FOCS 2020) showed that Ω(n) space is necessary to yield a 2/2+ε approximation of the optimum value; they also presented an algorithm that yields a 2/2−ε approximation of the optimum value using O(ε−2log⁡n) space.In this paper, we not only focus on approximating the optimum value, but also on obtaining the corresponding Boolean assignment using sublinear o(mn) space. We present randomized single-pass algorithms that w.h.p.1 yield:•A 1−ε approximation using O˜(n/ε3) space and exponential post-processing time•A 3/4−ε approximation using O˜(n/ε) space and polynomial post-processing time. Our ideas also extend to dynamic streams. However, we show that the streaming k-SAT problem, which asks whether one can satisfy all size-k input clauses, must use Ω(nk) space.We also consider the related minimum satisfiability problem (Min-SAT), introduced by Kohli et al. (SIAM J. Discrete Math. 1994), that asks to find an assignment that minimizes the number of satisfied clauses. For this problem, we give a O˜(n2/ε2) space algorithm, which is sublinear when m=ω(n), that yields an α+ε approximation where α is the approximation guarantee of the offline algorithm. If each variable appears in at most f clauses, we show that a 2fn approximation using O˜(n) space is possible.Finally, for the Max-AND-SAT problem where clauses are conjunctions of literals, we show that any single-pass algorithm that approximates the optimal value up to a factor better than 1/2 with success probability at least 2/3 must use Ω(mn) space.