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Packing Odd Walks and Trails in Multiterminal Networks

Authors Maxim Akhmedov , Maxim Babenko



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Maxim Akhmedov
  • Department of Mathematical Logic and Algorithms, Moscow State University, Russia
Maxim Babenko
  • Higher School of Economics, Moscow, Russia

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Maxim Akhmedov and Maxim Babenko. Packing Odd Walks and Trails in Multiterminal Networks. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 5:1-5:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.STACS.2023.5

Abstract

Let G be an undirected network with a distinguished set of terminals T ⊆ V(G) and edge capacities cap: E(G) → ℝ_+. By an odd T-walk we mean a walk in G (with possible vertex and edge self-intersections) connecting two distinct terminals and consisting of an odd number of edges. Inspired by the work of Schrijver and Seymour on odd path packing for two terminals, we consider packings of odd T-walks subject to capacities cap. First, we present a strongly polynomial time algorithm for constructing a maximum fractional packing of odd T-walks. For even integer capacities, our algorithm constructs a packing that is half-integer. Additionally, if cap(δ(v)) is divisible by 4 for any v ∈ V(G)-T, our algorithm constructs an integer packing. Second, we establish and prove the corresponding min-max relation. Third, if G is inner Eulerian (i.e. degrees of all nodes in V(G)-T are even) and cap(e) = 2 for all e ∈ E, we show that there exists an integer packing of odd T-trails (i.e. odd T-walks with no repeated edges) of the same value as in case of odd T-walks, and this packing can be found in polynomial time. To achieve the above goals, we establish a connection between packings of odd T-walks and T-trails and certain multiflow problems in undirected and bidirected graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Network optimization
Keywords
  • Odd path
  • signed and bidirected graph
  • multiflow
  • polynomial algorithm

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References

  1. Maxim A. Babenko and Alexander V. Karzanov. Free multiflows in bidirected and skew-symmetric graphs. Discret. Appl. Math., 155(13):1715-1730, 2007. Google Scholar
  2. B. V. Cherkassky. A solution of a problem on multicommodity flows in a network. Ekonomika i Matematicheskie Metody, 13(1):143-151, 1977. Google Scholar
  3. M. Chudnovsky, J. Geelen, and W. Cunningham. An algorithm for packing non-zero A-paths in group-labelled graphs. Combinatorica, 28(2):145-161, 2008. Google Scholar
  4. Ross Churchley, Bojan Mohar, and Hehui Wu. Weak duality for packing edge-disjoint odd (u, v)-trails. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 2086-2094. SIAM, 2016. Google Scholar
  5. Sharat Ibrahimpur and Chaitanya Swamy. Min-max theorems for packing and covering odd (u, v)-trails. In Friedrich Eisenbrand and Jochen Könemann, editors, Integer Programming and Combinatorial Optimization - 19th International Conference, IPCO 2017, Waterloo, ON, Canada, June 26-28, 2017, Proceedings, volume 10328 of Lecture Notes in Computer Science, pages 279-291. Springer, 2017. Google Scholar
  6. Alexander Karzanov. On a class of maximum multicommodity flow problems with integer optimal solutions. American Mathematical Society Translations, Ser. 2, 158:81-99, February 1994. URL: https://doi.org/10.1090/trans2/158/09.
  7. L. Lovász. On some connectivity properties of Eulerian graphs. Acta Math. Akad. Sci. Hung., 28:129-138, 1976. Google Scholar
  8. W. Mader. Über die maximalzahl kantendisjunkter H-wege. Archiv der Mathematik (Basel), 31:382-402, 1978. Google Scholar
  9. G. Pap. Packing non-returning A-paths algorithmically. Discrete Mathematics, 308(8):1472-1488, 2008. Google Scholar
  10. Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer Science & Business Media, 2003. Google Scholar
  11. Alexander Schrijver and Paul D. Seymour. Packing odd paths. J. Comb. Theory, Ser. B, 62(2):280-288, 1994. Google Scholar
  12. Yutaro Yamaguchi. Combinatorial Optimization on Group-Labeled Graphs. PhD thesis, Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Japan, 2016. Google Scholar
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