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Spanning trees short or small

Abstract

We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number kk of nodes are required to be connected in the solution. A prototypical example is the kkMST problem in which we require a tree of minimum weight spanning at least kk nodes in an edge-weighted graph. We show that the kkMST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio 2k2\sqrt{k} for the general edge-weighted case and O(k1/4)O(k^{1/4}) for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding kk-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.Comment: 27 page

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