On the Approximability of Time Disjoint Walks
We introduce the combinatorial optimization problem Time Disjoint Walks. This problem takes as input a digraph \(G\) with positive integer arc lengths, and \(k\) pairs of vertices that each represent a trip demand from a source to a destination. The goal is to find a path and delay for each demand so that no two trips occupy the same vertex at the same time, and so that the sum of trip times is minimized. We show that even for DAGs with max degree \(\varDelta \le 3\), Time Disjoint Walks is APX-hard. We also present a natural approximation algorithm, and provide a tight analysis. In particular, we prove that it achieves an approximation ratio of \(\varTheta (k/\log k)\) on bounded-degree DAGs, and \(\varTheta (k)\) on DAGs and bounded-degree digraphs.
KeywordsHardness of approximation Approximation algorithms Disjoint Paths problem
- 7.Kleinberg, J.M.: Approximation algorithms for disjoint paths problems. Ph.D. thesis. Massachusetts Institute of Technology (1996)Google Scholar
- 11.Scheffler, P.: A practical linear time algorithm for disjoint paths in graphs with bounded tree-width. TU, Fachbereich 3 (1994)Google Scholar
- 14.Srinivas, A., Modiano, E.: Minimum energy disjoint path routing in wireless ad-hoc networks. In: Proceedings of the 9th Annual International Conference on Mobile Computing and Networking, pp. 122–133. ACM (2003)Google Scholar
- 16.Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. In: Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, pp. 681–690. ACM (2006)Google Scholar