# Computing disjoint paths with length constraints

## Abstract

We show that the problem of computing a pair of disjoint paths between nodes *s* and *t* of an undirected graph, each having at most *K, K ε Z*^{+}, edges is NP-complete. A heuristic for its optimization version is given whose performance is within a constant factor from the optimal. It can be generalized to compute any constant number of disjoint paths. We also generalize an algorithm in [1] to compute the maximum number of edge disjoint paths of the shortest possible length between *s* and *t*. We show that it is NP-complete to decide whether there exist at least *K, K ε Z*^{+}, disjoint paths that may have at most *S*+1 edges, where *S* is the minimum number of edges on any path between *s* and *t*. In addition, we examine a generalized version of the problem where disjoint paths are routed either between a node pair (*s*1, *t*1) or a node pair (*s*2, *t*2). We show that it is NP-hard to find the maximum number of disjoint paths that either connect pair (*s*1, *t*1) the shortest way or (*s*2, *t*2) the shortest way.

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## References

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