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Path packing and a related optimization problem

Author

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  • Natalia Vanetik

    (Ben-Gurion University)

Abstract

Let G be a supply graph, with the node set N and edge set E, and (T,S) be a demand graph, with T⊆N, S∩E=∅. Observe paths whose end-vertices form pairs in S (called S-paths). The following path packing problem for graphs is fundamental: what is the maximal number of S-paths in G? In this paper this problem is studied under two assumptions: (a) the node degrees in N∖T are even, and (b) any three distinct pairwise intersecting maximal stable sets A,B,C of (T,S) satisfy A∩B=B∩C=A∩C (this condition was defined by A. Karzanov in Linear Algebra Appl. 114–115:293–328, 1989). For any demand graph violating (b) the problem is known to be NP-hard even under (a), and only a few cases satisfying (a) and (b) have been solved. In each of the solved cases, a solution and an optimal dual object were defined by a certain auxiliary “weak” multiflow optimization problem whose solutions supply constructive elements for S-paths and concatenate them into an S-path packing by a kind of matching. In this paper the above approach is extended to all demand graphs satisfying (a) and (b), by proving existence of a common solution of the S-path packing and its weak counterpart. The weak problem is very interesting for its own sake, and has connections with such topics as Mader’s edge-disjoint path packing theorem and b-factors in graphs.

Suggested Citation

  • Natalia Vanetik, 2009. "Path packing and a related optimization problem," Journal of Combinatorial Optimization, Springer, vol. 17(2), pages 192-205, February.
    • Handle: RePEc:spr:jcomop:v:17:y:2009:i:2:d:10.1007_s10878-007-9107-z
    DOI: 10.1007/s10878-007-9107-z
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    Cited by:

    1. Natalia Vanetik, 2012. "On the fractionality of the path packing problem," Journal of Combinatorial Optimization, Springer, vol. 24(4), pages 526-539, November.

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