On Tree-Preserving Constraints

Abstract

Tree convex constraints are extensions of the well-known row convex constraints. Just like the latter, every path-consistent tree convex constraint network is globally consistent. This paper studies and compares three subclasses of tree convex constraints which are called chain-, path-and tree-preserving constraints respectively. While the tractability of the subclass of chain-preserving constraints has been established before, this paper shows that every chain-or path-preserving constraint network is in essence the disjoint union of several independent connected row convex constraint networks, and hence (re-)establish the tractability of these two subclasses of tree convex constraints. We further prove that, when enforcing arc-and path-consistency on a tree-preserving constraint network, in each step, the network remains tree-preserving. This ensures the global consistency of the tree-preserving network if no inconsistency is detected. Moreover, it also guarantees the applicability of the partial path-consistency algorithm to tree-preserving constraint networks, which is usually more efficient than the path-consistency algorithm for large sparse networks. As an application, we show that the class of tree-preserving constraints is useful in solving the scene labelling problem.

Exploring Directional Path-Consistency for Solving Constraint Networks

“…7. A CRC constraint is a special tree-preserving constraint where related tree domains are chains [21].…”

“…Theorem 2. [21] Let T i and T j be two nonempty tree domains and m i and m j their standard majority operations. Suppose R ij ⊆ T i × T j is a nonempty constraint such that both R ij and R ji are arc-consistent.…”

“…Several important constraint classes have been found to be majority-closed. The most well-known one is the class of connected row-convex (CRC) constraints [16], which is further generalized to a larger class of tree-preserving constraints [20]. The class of CRC constraints has been successfully applied to temporal reasoning [24], logical filtering [25], and geometric reasoning [23], and the class of tree-preserving constraints can model a large subclass of the scene labeling problem [20].…”

“…The most well-known one is the class of connected row-convex (CRC) constraints [16], which is further generalized to a larger class of tree-preserving constraints [20]. The class of CRC constraints has been successfully applied to temporal reasoning [24], logical filtering [25], and geometric reasoning [23], and the class of tree-preserving constraints can model a large subclass of the scene labeling problem [20]. We also conduct experimental evaluations to compare DPC * to the state-of-the-art algorithms for solving majority-closed constraints, and show that DPC * significantly outperforms the latter algorithms.…”

On tree-preserving constraints

“…This paper is a significant extension of the conference paper [26]. Apart from detailed proofs for most of our results and more figures for illustrating the key notions and techniques, we make the following major extensions: (i) We add Section 4 to show that enforcing PPC will enable us to find a solution for a tree-preserving constraint network in a backtrack-free style.…”