Graphs with variable edge costs: a model for scheduling a vehicle subject to speed and timing restraints

Consider a graph with a range of costs associated with each vertex and each edge of the graph. The author studies paths in such graphs where the cost of each edge of the path must be chosen within the range of costs for that edge and the sum of the chosen costs of the edges from the start of the path to any vertex in the path must belong to the range of allowable costs for that vertex. He shows that for a fixed sequence of vertices, a linear time algorithm solves the problem of choosing the edge costs to satisfy the vertex cost constraints along the path. The problem of deciding if such a path exists from a given vertex to another specified vertex (without specifying the intermediate vertices on the path) is NP-complete. The problem of finding a tour of the vertices satisfying the cost constraints is also shown to be NP-complete. The study of these graphs is motivated by problems concerned with the routing of a robot vehicle with upper and lower speed limits and timing constraints for reaching each workcell