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 $$\varOmega (\sqrt{n})$$ space is necessary to yield a $$\sqrt{2}/2+\varepsilon $$ approximation of the optimum value; they also presented an algorithm that yields a $$\sqrt{2}/2-\varepsilon $$ approximation of the optimum value using $$O(\varepsilon ^{-2}\log n)$$ space. In this paper, we focus not only 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. (W.h.p. denotes “with high probability”. Here, we consider $$1-1/{{\,\textrm{poly}\,}}(n)$$ or $$1-1/{{\,\textrm{poly}\,}}(m)$$ as high probability.) yield: Our ideas also extend to dynamic streams. On the other hand, we show that the streaming k-SAT problem that asks to decide whether one can satisfy all size-k input clauses must use $$\varOmega (n^k)$$ space. We also consider the related minimum satisfiability problem ( $$\text {Min-SAT} $$ ), 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 $$\tilde{O}(n^2/\varepsilon ^2)$$ space algorithm, which is sublinear when $$m = \omega (n)$$ , that yields an $$\alpha +\varepsilon $$ approximation where $$\alpha $$ is the approximation guarantee of the offline algorithm. If each variable appears in at most f clauses, we show that a $$2\sqrt{fn}$$ approximation using $$\tilde{O}(n)$$ 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 $$\varOmega (mn)$$ space.