On The Power of Tree Projections: Structural Tractability of Enumerating CSP Solutions

TL;DR

This paper explores the structural conditions under which enumerating solutions to CSPs can be done efficiently with polynomial delay, using tree projections to generalize existing decomposition methods.

Contribution

It introduces a unified framework based on tree projections for analyzing the tractability of enumerating CSP solutions, extending known results to broader classes.

Findings

Polynomial delay enumeration is achievable for certain classes of CSPs.

Tree projection-based decompositions generalize existing methods.

Dichotomies are established for classes with bounded arity.

Abstract

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 general framework based on the notion of tree projection of hypergraphs is considered, which generalizes all known decomposition methods. 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. These results are shown to be tight, by exhibiting dichotomies for classes of structures having bounded arity and where the tree decomposition method is considered.