Graph-DTP: Graph-Based Algorithm for Solving Disjunctive Temporal Problems

We study an expressive quantitative temporal model: disjunctive temporal problem (DTP), which was first proposed only in 1998 (Stergiou and Koubarakis). As extension of temporal constraint satisfaction problem (TCSP) (Dechter et al. 1991), DTP differs from TCSP in that two disjuncts in a same disjunctive constraint do not necessarily refer to same temporal variables. Traditionally, most of the DTP algorithms in the literature solve DTPs by treating them as constraint satisfaction problems (CSPs), and searching for solutions using standard CSP techniques, e.g. backtracking, back-jumping, forward checking, semantic branching, removal of subsumed variables, nogood recording, etc. Those CSP techniques are powerful in solving DTPs. However, an evident drawback of viewing DTPs as general CSPs is that much semantic information encoded in DTPs is neglected. In fact we can mine rich semantic information that can be exploited to reduce search space for DTPs (more than semantic branching). Through some topological analysis on the graphical representation of the problems, some techniques are developed to help to search solutions for other temporal models (e.g. TCSP), or to identify "crucial subproblems" for CSP (Epstein and Wallace, 2006). However, little effort has been made to exploit the inherent topological information in solving DTPs. Our idea runs on a graphical representation of DTPs - disjunctive temporal network (DTN). We define DTN as an edge-labeled weighted digraph, on which some relevant concepts are identified. Then, we define the concept of equivalency between DTNs with respect to their consistency. For a given DTN, deciding its consistency is ascribed to check the consistency of a DTN which is equivalent to it and has less constraints (edges). We iteratively reduce a DTN to a simpler but equivalent one according to a set of designed reduction rules (which can be performed within polynomial time). It is hoped that when the DTN reaches a fixed point under such red- - uction operation, the resulted DTN has minimal edges (which is like backdoor in SAT, or "near clique"). At last the resulted DTN (DTP) is transferred to CSP search phase, where we derive a special variable ordering strategy again through the DTN structure. We shall describe the generation of DTN structure, the DTN reduction rules, the implementation of the complete graph-DTP algorithm, and some first results of this approach.