A Network-Flow Technique for Finding Low-Weight Bounded-Degree Spanning Trees

Given a graph with edge weights satisfying the triangle inequality, and a degree bound for each vertex, the problem of computing a low weight spanning tree such that the degree of each vertex is at most its specified bound is considered. In particular, modifying a given spanning tree T using adoptions to meet the degree constraints is considered. A novel network-flow based algorithm for finding a good sequence of adoptions is introduced. The method yields a better performance guarantee than any previous algorithm. If the degree constraint d(v) for each v is at least 2, the algorithm is guaranteed to find a tree whose weight is at most the weight of the given tree times 2 − min { d(v)−2 degT (v)−2 : degT (v) > 2 } , where degT (v) is the initial degree of v. Equally importantly, it takes this approach to the limit in the following sense: if any performance guarantee that is solely a function of the topology and edge weights of a given tree holds for any algorithm at all, then it also holds for the given algorithm. Examples are provided in which no lighter tree meeting the degree constraint exists. Linear-time algorithms are provided with the same worst-case performance guarantee. Choosing T to be a minimum spanning tree yields approximation algorithms with factors less than 2 for the general problem on geometric graphs with distances induced by various Lp norms. Finally, examples of Euclidean graphs are provided in which the ratio of the lengths of an optimal Traveling Salesman path and a minimum spanning tree can be arbitrarily close to 2. A preliminary version of this paper appeared in the Proceedings of the 5th International Integer Programming and Combinatorial Optimization Conference (IPCO), June 1996, pages 105-117. Center for Parallel Computing, Universität zu Köln, D-50923 Köln, Germany. E-Mail: [email protected]. Computer Science Department and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742. Research supported by NSF Research Initiation Award CCR-9307462 and an NSF CAREER Award CCR-9501355. E-mail : [email protected]. Center for Parallel Computing, Universität zu Köln, D-50923 Köln, Germany. E-Mail: [email protected]. Department of Computer Science, The University of Texas at Dallas, Box 830688, Richardson, TX 75083-0688. Research supported in part by NSF Research Initiation Award CCR-9409625. E-mail : [email protected]. Dept. of Computer Science, Dartmouth College, Hanover NH 03755-3510. Part of this research was done while at School of ORIE, Cornell University, Ithaca NY 14853 and supported by Éva Tardos’ NSF PYI grant DDM-9157199. E-mail : [email protected].