Spanning trees short or small

TL;DR

This paper investigates the complexity and approximation algorithms for the k-MST problem and related short network problems, providing new algorithms, exact solutions for specific graph classes, and insights into problem hardness.

Contribution

It introduces approximation algorithms for k-MST with performance ratios depending on k, and presents polynomial-time solutions for certain graph classes and boundary points, advancing understanding of small tree problems.

Findings

k-MST is NP-hard in Euclidean plane

Approximation algorithms with ratios 2√k and O(k^{1/4})

Polynomial solutions for decomposable graphs and boundary points

Abstract

We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ 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 $k$-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.