Pseudo Polynomial-Time Top-k Algorithms for d-DNNF Circuits

We are interested in computing k most preferred models of a given d-DNNF circuit C, where the preference relation is based on an algebraic structure called a monotone, totally ordered, semigroup (K, otimes, <). In our setting, every literal in C has a value in K and the value of an assignment is an element of K obtained by aggregating using otimes the values of the corresponding literals. We present an algorithm that computes k models of C among those having the largest values w.r.t. <, and show that this algorithm runs in time polynomial in k and in the size of C. We also present a pseudo polynomial-time algorithm for deriving the top-k values that can be reached, provided that an additional (but not very demanding) requirement on the semigroup is satisfied. Under the same assumption, we present a pseudo polynomial-time algorithm that transforms C into a d-DNNF circuit C' satisfied exactly by the models of C having a value among the top-k ones. Finally, focusing on the semigroup (mathbbN, +, <), we compare on a large number of instances the performances of our compilation-based algorithm for computing k top solutions with those of an algorithm tackling the same problem, but based on a partial weighted MaxSAT solver.