Cost-based Filtering for Shorter Path Constraints

Abstract

Many real world problems, e.g. personnel scheduling and transportation planning, can be modeled naturally as Constrained Shortest Path Problems (CSPPs), i.e., as Shortest Path Problems with additional constraints. A well studied problem in this class is the Resource Constrained Shortest Path Problem. Reduction techniques are vital ingredients of solvers for the CSPP, that is frequently NP-hard, depending on the nature of the additional constraints. Viewed as heuristics, these techniques have not been studied theoretically with respect to their efficiency, i.e., with respect to the relation of filtering power and running time. Using the concepts of Constraint Programming, we provide a theoretical study of cost-based filtering for shorter path constraints on acyclic, on undirected, and on directed graphs that do not contain negative cycles. We then show empirically how reasoning about path-substructures in combination with CP-based Lagrangian relaxation can help to improve significantly over previously developed problem-tailored filtering algorithms for the resource constrained shortest path problem and investigate the impact of required-edge detection, undirected versus directed filtering, and the choice of the algorithm optimizing the Lagrangian dual.

Revisiting the tree Constraint

“…al. introduced the DomReachability constraint [11,10,16] for solving path partitioning problems. It uses a similar graph variable description [4] .Their constraint maintains structural relationships between a flow graph, its dominator tree and its transitive closure.…”

“…This cost is due to the algorithm used for maintaining the transitive closure, which is not necessary for tree partitioning. The first algorithm [11] consisted in performing one DFS per node whereas the current algorithm [16] works on a reduced graph.…”

“…This is pretty obvious because the reachability property does not exclude cycles whereas tree properties do. Nevertheless they use DomReachability for path partitioning through the global constraint Simple Path with Mandatory Nodes (SPMN) [11,10,16]. As tree partitioning is a polynomial relaxation of path partitioning, a complete filtering over tree partitioning could be expected.…”

“…In the same way, difficult problems modeled and solved thanks to graph theory have been successfully tackled in constraint programming and, more particularly, thanks to global constraints. This mainly consists of constraints around graph and subgraph isomorphism [17,19], search paths in graphs [11,10,16], even minimum cost spanning trees [14] and graph partitioning constraints like the tree constraint [2] Such a constraint is mainly involved in practical applications like vehicle routing, mission planning, DNA sequencing, or phylogeny.…”

Combining Two Structured Domains for Modeling Various Graph Matching Problems

“…Various constraints have been defined over such graph variables (or their preceding specialized models); see for instance the cycle [18], tree [21], path [22,23], minimum spanning tree [24] or spanning tree optimization constraint [25]. In the remainder of this article, we only use the two simple constraints Subgraph(G 1 , G 2 ) (also denoted…”