Spanning trees short or small
R. Ravi, R. Sundaram, Madhav V. Marathe, S. S. Ravi, Daniel J., Rosenkrantz

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 of nodes are required to be connected in the solution. A prototypical example is the MST problem in which we require a tree of minimum weight spanning at least nodes in an edge-weighted graph. We show that the MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio for the general edge-weighted case and 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…
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Taxonomy
TopicsVLSI and FPGA Design Techniques · Advanced Graph Theory Research · Interconnection Networks and Systems
