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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
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 [8, vol. B] for more information on polymatroids.
References
Bansal, N., Khandekar, R., Könemann, J., Nagarajan, V., Peis, B.: On generalizations of network design problems with degree bounds. Math. Program. Ser. A 141, 479–506 (2013)
Bansal, N., Khandekar, R., Nagarajan, V.: Additive guarantees for degree-bounded directed network design. SIAM J. Comput. 39(4), 1413–1431 (2009)
Lau, L.C., Ravi, R., Singh, M.: Iterative Methods in Combinatorial Optimization, vol. 46. Cambridge University Press, Cambridge (2011)
McDiarmid, C.J.H.: Rado’s theorem for polymatroids. Math. Proc. Cambridge Philos. Soc. 78, 263–281 (1975)
Molnár, M., Durand, S., Merabet, M.: Approximation of the degree-constrained minimum spanning hierarchies. In: Halldórsson, M.M. (ed.) SIROCCO 2014. LNCS, vol. 8576, pp. 96–107. Springer, Heidelberg (2014)
Molnár, M., Durand, S., Merabet, M.: A new formulation of degree-constrained spanning problems. In: Proceedings of 9th International Colloquium on Graph Theory and Combinatorics (ICGT) (2014). http://oc.inpg.fr/conf/icgt2014/
Molnár, M., Newman, A., Sebő, A.: Travelling salesmen on bounded degree trails, Hausdorff report (in preparation) (2015)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics, vol. 24. Springer, Heidelberg (2003)
Sebő, A.: Travelling salesmen on bounded degree trails. In: Presentation at HIM Connectivity Workshop in Bonn, Presentation (2015). https://www.youtube.com/watch?v=5Do2JMhgrCM
Singh, M., Lau, L.C.: Approximating minimum bounded degree spanning trees to within one of optimal. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC), pp. 661–670 (2007)
Zenklusen, R.: Matroidal degree-bounded minimum spanning trees. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1512–1521 (2012)
Acknowledgements
We are grateful to Michel Goemans, Anupam Gupta, Neil Olver, and András Sebő for inspiring discussions, and to the anonymous referees 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 to the generous support and inspiring environment provided by the HIM and the organizers of the trimester program.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Singh, M., Zenklusen, R. (2016). k-Trails: Recognition, Complexity, and Approximations. In: Louveaux, Q., Skutella, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 2016. Lecture Notes in Computer Science(), vol 9682. Springer, Cham. https://doi.org/10.1007/978-3-319-33461-5_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-33461-5_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-33460-8
Online ISBN: 978-3-319-33461-5
eBook Packages: Computer ScienceComputer Science (R0)