Structural tractability of enumerating CSP solutions

The problem of deciding whether CSP instances admit solutions has been deeply studied in the literature, and several structural tractability results have been derived so far. However, constraint satisfaction comes in practice as a computation problem where the focus is either on finding one solution, or on enumerating all solutions, possibly projected to some given set of output variables. The paper investigates the structural tractability of the problem of enumerating (possibly projected) solutions, where tractability means here computable with polynomial delay (WPD), since in general exponentially many solutions may be computed. A framework based on the notion of tree projection of hypergraphs is considered, which generalizes all structural decomposition methods that are based on decomposing a given instance into suitable tree-like groups of polynomial-time computable subproblems. Tractability results have been obtained both for classes of structures where output variables are part of their specification, and for classes of structures where computability WPD must be ensured for any possible set of output variables. By exhibiting dichotomies, these results are shown to be tight for classes of structures having bounded arity.

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