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Approximating Spanning Trees with Few Branches

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Abstract

Given an undirected, connected graph, the aim of the minimum branch-node spanning tree problem is to find a spanning tree with the minimum number of nodes of degree larger than 2. The problem is motivated by network design problems where junctions are significantly more expensive than simple end- or through-nodes, and are thus to be avoided. Unfortunately, it is NP-hard to recognize instances that admit an objective value of zero, rendering the search for guaranteed approximation ratios futile.

We suggest to investigate a complementary formulation, called maximum path-node spanning tree, where the goal is to find a spanning tree that maximizes the number of nodes with degree at most two. While the optimal solutions (and the practical applications) of both formulations coincide, our formulation proves more suitable for approximation. In fact, it admits a trivial 1/2-approximation algorithm. Our main contribution is a local search algorithm that guarantees a ratio of 6/11, as well as showing that the problem is APX-hard, i.e., it does not allow a PTAS.

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Notes

  1. The term forward node is motivated by the technical background of WDM networks, and, e.g., consistent with the definition of maximum forward-node spanning trees [17].

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Authors and Affiliations

  1. Institute of Computer Science, Friedrich-Schiller-University, Jena, Germany

    Markus Chimani

  2. Institute of Computer Science, University of Würzburg, Würzburg, Germany

    Joachim Spoerhase

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  1. Markus Chimani
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  2. Joachim Spoerhase
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Correspondence to Joachim Spoerhase.

Additional information

This research started from an open problem posed at Dagstuhl Seminar 11191. The authors are grateful for Dagstuhl’s support.

Markus Chimani was funded by a Carl-Zeiss-Foundation juniorprofessorship

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Chimani, M., Spoerhase, J. Approximating Spanning Trees with Few Branches. Theory Comput Syst 56, 181–196 (2015). https://doi.org/10.1007/s00224-014-9556-6

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