Finding paths with the right cost

We study a problem related to finding shortest paths in weighted graphs. We ask whether or not there is a path between two nodes that is of a given cost. The edge weights of the graph can be both positive and negative integers, or even integer vectors. We show that most variants of this problem are NP-complete. We also develop a pseudopolynomial algorithm for the case where the edge weights are integers. The running time of this algorithm is O((MN3)-N-2 + \w\min(\w\, M)N-2) where N is the number of nodes in the graph, M is the largest absolute Value of any edge weight, and ur is the target cost. The algorithm is based on preprocessing the graph with a relaxation algorithm to eliminate the effects of weight sign alternations along a path.