On tree-preserving constraints

Abstract

The study of tractable subclasses of constraint satisfaction problems is a central topic in constraint solving. 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. However, it is NP-complete to decide whether a tree convex constraint network has solutions. This paper studies and compares three subclasses of tree convex constraints, which are called chain-, path-, and tree-preserving constraints respectively. The class of tree-preserving constraints strictly contains the subclasses of path-preserving and arc-consistent chain-preserving constraints. We prove that, when enforcing strong pathconsistency on a tree-preserving constraint network, in each step, the network remains treepreserving. This ensures the global consistency of consistent tree-preserving networks after enforcing strong path-consistency, and also guarantees the applicability of the partial pathconsistency algorithms to tree-preserving constraint networks, which is usually much more efficient than the path-consistency algorithms for large sparse constraint 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]. Definition 10.…”

“…Let Q k be the restriction of M (k) to {v 1 , v 2 , ..., v n−k } (0 ≤ k < n). In essence, Q k is obtained by eliminating v n−k+1 from Q k−1 (Lines 2-5 or Lines [11][12][13][14][15][16][17][18][19][20], while also enforcing AC (Lines 7-10) for v n−k+1 relative to all its successors if Algorithm 2: DPC * Input : A binary constraint network N = V, D, C ; an ordering ≺ = (v1, . .…”

“…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.…”