Structural Tractability of Constraint Optimization

Abstract

Several variants of the Constraint Satisfaction Problem have been proposed and investigated in the literature for modeling those scenarios where solutions are associated with costs. Within these frameworks, computing an optimal solution (short: Min problem), enumerating the best K solutions (Top- K), and computing the next solution following one that is given at hand (Next) are all $\mbox{\rm NP}$-hard problems. In fact, only some restricted islands of tractability for them have been singled out in the literature. The paper fills the gap, by studying the complexity of Min, Top- K, and Next over classes of acyclic and nearly acyclic instances, as they can be identified via structural decomposition methods. The analysis is provided for both monotone cost-functions and non-monotone ones (which have been largely ignored so far). Also, multi-criteria optimization is considered, as instances may have a number of cost functions to be minimized together, according to a given precedence relationship. Large islands of tractability are identified and, for classes of bounded-arity instances, the tractability frontier of constraint optimization is precisely charted.