Consistency Methods for Temporal Reasoning

1. Polynomial number of polynomial-size ternary constraints One global, exponential size constraint Consistency Methods for Temporal Reasoning Lin Xu and Berthe Y. Choueiry Constraint Systems Laboratory • Department of Computer Science and Engineering, University of Nebraska-Lincoln • {lxu|choueiry}@cse.unl.edu • We propose new efficient algorithms for solving Simple Temporal Problems (STPs) and Temporal Constraint Satisfaction Problems (TCSPs). Our contributions are: • Answers to three research questions: • Is there a better algorithm than Floyd-Warshall (F-W) to ensure consistency and minimality of the STP? • Is there an efficient consistency algorithm to reduce the size of the TCSP? • How can we improve the performance of BT-TCSP, an exponential time procedure for solving the TCSP? • Design of new generators for random STPs and TCSPs that guarantee the existence of at least one solution. • Evaluation & empirical evidence on benchmarks on random problems Solving TCSPs is an NP-hard problem. It requires search (BT-TCSP) Formulate TCSP as a meta-CSP Find all solutions to the meta-CSP For STPs, there are efficient procedures for determining: • Consistency: F-W, DPC [2], PPC [1]. • Minimality: F-W & PPC [1]. 1. Is there a better algorithm for solving STP than the Floyd-Warshall algorithm (F-W)? We propose STP a new closure algorithm for solving STPs, an adaptation of PPC [1] to temporal networks. • Two parameters for improvements: • Exploiting the topology the graph:decompose the graph according into its bi-connected component using the articulation points (AP), and solve each component independently (cost bounded by size of the largest component). Known technique, but never tested empirically • Exploiting the semantics of the temporal constraints: which are convex. • Partial path consistency (PPC): PPC is a closure algorithm,applicable to general CSPs. It requires triangulation of constraint graph. In general, PPC does not result in the minimal network. But for convex constraints, it guarantees minimality (just like F-W, but much cheaper in practice). • STP improves on PPC: • Operates on the constraint graph as if composed of triangles, not edges • Automatically decomposes the constraint graph into bi-connected components • Always cheaper than PPC and F-W • Best known algorithm for solving STP. Edge Ordering (EdgeOrd): We order the edges using ‘triangle adjacency’ property. The priority list is a by-product of triangulation. EdgeOrd has two advantages: (1) localized backtracking (2) automatic decomposition of the constraint graph. 2. Is there an efficient algorithm for filtering the TCSP? We propose AC, a polynomial-time algorithm that approximates generalized arc-consistency for the TCSP, which is NP-hard. AC: The goal of the AC algorithm is to remove inconsistent intervals from the domain of the variables of the meta-CSP, as a preprocessing step to search. The meta-CSP has a unique, global constraint of exponential size. Defining this constraint requires solving the meta-CSP, which is NP-hard. We use polynomial number of polynomial-size ternary constraints instead of this global constraint. An in a given interval that does not satisfy any ternary constraints cannot possibly satisfy the global constraint. Hence, AC safely can removed from the domain of edge. Validation by empirical evaluations We test the following combinations of TCSP-solvers. Experimental results confirm the significance of our approach. The best TCSP-solver is STP-TCSP (based on STP and using AC, EdgeOrd & NewCyc). It outperforms the old TCSP-solver (DPC-TCSP) by a factor 500 (median) in the number of constraint checks. 3. How can we improve performance of BT-TCSP? We propose: • to exploit the existence of articulation points (again!) • to eliminate unnecessary STP-consistency checks (NewCyc) • a good variable ordering heuristic in BT-TCSP (EdgeOrd) New cycle check: checks presence of new cycles, O (|E |). We check consistency of STP only if a new cycle is added. NewCyc restricts update to the newly formed bi-connected component, reducing #CC. Directions for future research • Ensure optimality of AC • Use AC in a lookahead strategy • Use dynamic bundling to allow BT-TCSP to work on large problems. References [1] Ch. Bliek and D. Sam-Haroud. Path Consistency for Triangulated Constraint Graphs. In Proc. of the 16th IJCAI , pages 456-461, Stockholm, Sweden, 1999  [2] R. Dechter, I. Meiri, and J. Pearl. Temporal Constraint Networks. Artificial Intelligence,. 49:61-95, 1991 [3] L. Xu & B.Y. Choueiry, A New Efficient Algorithm for Solving the Simple Temporal Problem, to appear, TIME 03. [4] L. Xu, Consistency Methods for Temporal Reasoning, MS thesis, ConSystLab, UNL, 2003. This work is supported by a grant from NASA-Nebraska, NSF CAREER Award #0133568, and a gift from Honeywell Laboratories.