Longest path problem

About: Longest path problem is a research topic. Over the lifetime, 3264 publications have been published within this topic receiving 102814 citations.

A potential field approach to path planning

Abstract: A path-planning algorithm for the classical mover's problem in three dimensions using a potential field representation of obstacles is presented. A potential function similar to the electrostatic potential is assigned to each obstacle, and the topological structure of the free space is derived in the form of minimum potential valleys. Path planning is done at two levels. First, a global planner selects a robot's path from the minimum potential valleys and its orientations along the path that minimize a heuristic estimate of the path length and the chance of collision. Then, a local planner modifies the path and orientations to derive the final collision-free path and orientations. If the local planner fails, a new path and orientations are selected by the global planner and subsequently examined by the local planner. This process is continued until a solution is found or there are no paths left to be examined. The algorithm solves a much wider class of problems than other heuristic algorithms and at the same time runs much faster than exact algorithms (typically 5 to 30 min on a Sun 3/260). >

An exact algorithm for the elementary shortest path problem with resource constraints: Application to some vehicle routing problems

Abstract: In this article, we propose a solution procedure for the Elementary Shortest Path Problem with Resource Constraints (ESPPRC). A relaxed version of this problem in which the path does not have to be elementary has been the backbone of a number of solution procedures based on column generation for several important problems, such as vehicle routing and crew pairing. In many cases relaxing the restriction of an elementary path resulted in optimal solutions in a reasonable computation time. However, for a number of other problems, the elementary path restriction has too much impact on the solution to be relaxed or might even be necessary. We propose an exact solution procedure for the ESPPRC, which extends the classical label correcting algorithm originally developed for the relaxed (nonelementary) path version of this problem. We present computational experiments of this algorithm for our specific problem and embedded in a column generation scheme for the classical Vehicle Routing Problem with Time Windows. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(3), 216–229 2004

An analysis of stochastic shortest path problems

Abstract: We consider a stochastic version of the classical shortest path problem whereby for each node of a graph, we must choose a probability distribution over the set of successor nodes so as to reach a certain destination node with minimum expected cost. The costs of transition between successive nodes can be positive as well as negative. We prove natural generalizations of the standard results for the deterministic shortest path problem, and we extend the corresponding theory for undiscounted finite state Markovian decision problems by removing the usual restriction that costs are either all nonnegative or all nonpositive.

Shortest-path and minimum-delay algorithms in networks with time-dependent edge-length

Abstract: In this paper the shortest-path problem in networks in which the delay (or weight) of the edges changes with time according to arbitrary functions is considered. Algorithms for finding the shortest path and minimum delay under various waiting constraints are presented and the properties of the derived path are investigated. It is shown that if departure time from the source node is unrestricted, then a shortest path can be found that is simple and achieves a delay as short as the most unrestricted path. In the case of restricted transit, it is shown that there exist cases in which the minimum delay is finite, but the path that achieves it is infinite.

Hamilton Paths in Grid Graphs

Abstract: A grid graph is a node-induced finite subgraph of the infinite grid. It is rectangular if its set of nodes is the product of two intervals. Given a rectangular grid graph and two of its nodes, we give necessary and sufficient conditions for the graph to have a Hamilton path between these two nodes. In contrast, the Hamilton path (and circuit) problem for general grid graphs is shown to be NP-complete. This provides a new, relatively simple, proof of the result that the Euclidean traveling salesman problem is NP-complete.