A Scalable Approximate Model Counter

Abstract

Abstract.Propositional model counting (#SAT), i.e., counting the number of satisfying assignments of a propositional formula, is a problem of significant theoretical and practical interest. Due to the inherent complexity of the problem, approximate model counting, which counts the number of satisfying assignments to within given tolerance and confidence level, was proposed as a practical alternative to exact model counting. Yet, approximate model counting has been studied essentially only theoretically. The only reported implementation of approximate model counting, due to Karp and Luby, worked only for DNF formulas. A few existing tools for CNF formulas are bounding model counters; they can handle realistic problem sizes, but fall short of providing counts within given tolerance and confidence, and, thus, are not approximate model counters.We present here a novel algorithm, as well as a reference implementation, that is the first scalable approximate model counter for CNF formulas. The algorithm works by issuing a polynomial number of calls to a SAT solver. Our tool, ApproxMC, scales to formulas with tens of thousands of variables. Careful experimental comparisons show that ApproxMC reports, with high confidence, bounds that are close to the exact count, and also succeeds in reporting bounds with small tolerance and high confidence in cases that are too large for computing exact model counts.

On Parallel Scalable Uniform SAT Witness Generation

“…The following result about Chernoff-Hoeffding bounds, proved in [5], plays an important role in the analysis of UniGen2.…”

“…Recently, Chakraborty et al proposed a new algorithm named UniGen [5], which improves upon the ideas of UniWit. In particular, UniGen provides stronger guarantees of uniformity by exploiting a deep connection between approximate counting and almost-uniform sampling [15].…”

“…Furthermore, UniGen has been shown to scale to formulae with hundreds of thousands of variables. Even so, UniGen is typically 2-3 orders of magnitude slower than a single call to a SAT solver and therefore, its runtime performance falls short of the performance of heuristic methods commonly employed in industry to generate stimuli for CRV 5 . In this paper, we offer several improvements to UniGen and obtain an algorithm with substantially improved performance that can be further scaled by parallelization to match the requirements of industry.…”

“…This is done along the same lines as in UniGen, which used an approximate model counter such as ApproxMC [5]. The procedure invokes a SAT solver through the function BSAT(φ, m, S).…”

“…In such scenarios, beginning with i = hashBits−1 in subsequent iterations saves considerable time. This heuristic is similar in spirit to "leapfrogging" in ApproxMC [5] and UniWit [4], but does not compromise the theoretical guarantees of UniGen2 in any way.…”