Mathematical Programming

, Volume 147, Issue 1–2, pp 81–123 | Cite as

Tree metrics and edge-disjoint \(S\)-paths

  • Hiroshi HiraiEmail author
  • Gyula Pap
Full Length Paper Series A


Given an undirected graph \(G=(V,E)\) with a terminal set \(S \subseteq V\), a weight function Open image in new window on terminal pairs, and an edge-cost \(a: E \rightarrow \mathbf{Z}_+\), the \(\mu \)-weighted minimum-cost edge-disjoint \(S\)-paths problem (\(\mu \)-CEDP) is to maximize \(\sum \nolimits _{P \in \mathcal{P}} \mu (s_P,t_P) - a(P)\) over all edge-disjoint sets \(\mathcal{P}\) of \(S\)-paths, where \(s_P,t_P\) denote the ends of \(P\) and \(a(P)\) is the sum of edge-cost \(a(e)\) over edges \(e\) in \(P\). Our main result is a complete characterization of terminal weights \(\mu \) for which \(\mu \)-CEDP is tractable and admits a combinatorial min–max theorem. We prove that if \(\mu \) is a tree metric, then \(\mu \)-CEDP is solvable in polynomial time and has a combinatorial min–max formula, which extends Mader’s edge-disjoint \(S\)-paths theorem and its minimum-cost generalization by Karzanov. Our min–max theorem includes the dual half-integrality, which was earlier conjectured by Karzanov for a special case. We also prove that \(\mu \)-EDP, which is \(\mu \)-CEDP with \(a = 0\), is NP-hard if \(\mu \) is not a truncated tree metric, where a truncated tree metric is a weight function represented as pairwise distances between balls in a tree. On the other hand, \(\mu \)-CEDP for a truncated tree metric \(\mu \) reduces to \(\mu '\)-CEDP for a tree metric \(\mu '\). Thus our result is best possible unless P = NP. As an application, we obtain a good approximation algorithm for \(\mu \)-EDP with “near” tree metric \(\mu \) by utilizing results from the theory of low-distortion embedding.

Mathematics Subject Classification

90C27 05C38 



We thank Alexander Karzanov for remarks on the earlier version of the paper, and thank the referees for helpful comments. The first author is supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and is partially supported by Aihara Project, the FIRST program from JSPS. The second author is supported by the Hungarian National Foundation for Scientific Research (OTKA) grant CK80124.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  1. 1.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyThe University of TokyoTokyoJapan
  2. 2.Department of Operations ResearchEötvös Loránd UniversityBudapestHungary

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