Auditable Algorithms for Approximate Model Counting

Abstract

The problem of model counting, i.e., counting satisfying assignments of a Boolean formula, is a fundamental problem in computer science, with diverse applications. Given #P-hardness of the problem, many algorithms have been developed over the years to provide an approximate model count. Recently, building on the practical success of SAT-solvers used as NP oracles, the focus has shifted from theory to practical implementations of such algorithms. This has brought to focus new challenges. In this paper, we consider one such challenge – that of auditable deterministic approximate model counters wherein a counter should also generate a certificate, which allows a user (often with limited computational power) to independently audit whether the count returned by an invocation of the algorithm is indeed within the promised bounds. We start by examining a celebrated approximate model counting algorithm due to Stockmeyer that uses polynomially many calls to a \Sigma^2_P oracle, and show that it can be audited via a \Pi^2_P formula on (n^2 log^2 n) variables, where n is the number of variables in the original formula. Since n is often large (10’s to 100’s of thousands) for typical instances, we ask if the count of variables in the certificate formula can be reduced – a critical question towards potential implementation. We show that this improvement in certification can be achieved with a tradeoff in the counting algorithm’s complexity. Specifically, we develop new deterministic approximate model counting algorithms that invoke a \Sigma^3_P oracle, but can be certified using a \Pi^2_P formula on fewer variables: our final algorithm uses just (n log n) variables. Our study demonstrates that one can simplify certificate checking significantly if we allow the counting algorithm to access a slightly more powerful oracle. We believe this shows for the first time how the audit complexity can be traded for the complexity of approximate counting.