Abstract
The notion of degree-constrained spanning hierarchies, also called k-trails, was recently introduced in the context of network routing problems. They describe graphs that are homomorphic images of connected graphs of degree at most k. First results highlight several interesting advantages of k-trails compared to previous routing approaches. However, so far, only little is known regarding computational aspects of k-trails. In this work we aim to fill this gap by presenting how k-trails can be analyzed using techniques from algorithmic matroid theory. Exploiting this connection, we resolve several open questions about k-trails. In particular, we show that one can recognize efficiently whether a graph is a k-trail, and every graph containing a k-trail is a \((k+1)\)-trail. Moreover, further leveraging the connection to matroids, we consider the problem of finding a minimum weight k-trail contained in a graph G. We show that one can efficiently find a \((2k-1)\)-trail contained in G whose weight is no more than the cheapest k-trail contained in G, even when allowing negative weights. The above results settle several open questions raised by Molnár, Newman, and Sebő.
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Notes
A polymatroid over a finite set N is a polytope \(P\subseteq {\mathbb {R}}^N_{\ge 0}\) described by \(P=\{x\in {\mathbb {R}}^N_{\ge 0} \mid x(S) \le f(S) \;\forall S\subseteq N\}\), where \(f:2^N \rightarrow {\mathbb {Z}}_{\ge 0}\) is a submodular function, and \(x(S) = \sum _{v\in S} x_v\). We refer the interested reader to [12, Volume B] for more information on polymatroids.
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Acknowledgements
We are grateful to Michel Goemans, Anupam Gupta, Neil Olver, and András Sebő for inspiring discussions, and to the anonymous reviewers for many helpful comments. This research project started while both authors were guests at the Hausdorff Research Institute for Mathematics (HIM) during the 2015 Trimester on Combinatorial Optimization. Both authors are very thankful for the generous support and inspiring environment provided by the HIM and the organizers of the trimester program.
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Rico Zenklusen was supported by Swiss National Science Foundation Grant 200021_165866.
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Singh, M., Zenklusen, R. k-Trails: recognition, complexity, and approximations. Math. Program. 172, 169–189 (2018). https://doi.org/10.1007/s10107-017-1113-z
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DOI: https://doi.org/10.1007/s10107-017-1113-z