To Encode or to Propagate? The Best Choice for Each Constraint in SAT

Sophisticated compact SAT encodings exist for many types of constraints. Alternatively, for instances with many (or large) constraints, the SAT solver can also be extended with built-in propagators (the SAT Modulo Theories approach, SMT). For example, given a cardinality constraint x 1 + … + x n  ≤ k, as soon as k variables become true, such a propagator can set the remaining variables to false, generating a so-called explanation clause of the form \(x_1 \wedge \ldots \wedge x_k \rightarrow{\bar{x_i}}\). But certain “bottle-neck” constraints end up generating an exponential number of explanations, equivalent to a naive SAT encoding, much worse than using a compact encoding with auxiliary variables from the beginning. Therefore, Abío and Stuckey proposed starting off with a full SMT approach and partially encoding, on the fly, only certain “active” parts of constraints. Here we build upon their work. Equipping our solvers with some additional bookkeeping to monitor constraint activity has allowed us to shed light on the effectiveness of SMT: many constraints generate very few, or few different, explanations. We also give strong experimental evidence showing that it is typically unnecessary to consider partial encodings: it is competitive to encode the few really active constraints entirely. This makes the approach amenable to any kind of constraint, not just the ones for which partial encodings are known.