Sequence Hypergraphs

  • Kateřina Böhmová
  • Jérémie Chalopin
  • Matúš Mihalák
  • Guido Proietti
  • Peter Widmayer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9941)

Abstract

We introduce sequence hypergraphs by extending the concept of a directed edge (from simple directed graphs) to hypergraphs. Specifically, every hyperedge of a sequence hypergraph is defined as a sequence of vertices (imagine it as a directed path). Note that this differs substantially from the standard definition of directed hypergraphs. Sequence hypergraphs are motivated by problems in public transportation networks, as they conveniently represent transportation lines. We study the complexity of some classic algorithmic problems, arising (not only) in transportation, in the setting of sequence hypergraphs. In particular, we consider the problem of finding a shortest st-hyperpath: a minimum set of hyperedges that “connects” (allows to travel to) t from s; finding a minimum st-hypercut: a minimum set of hyperedges whose removal “disconnects” t from s; or finding a maximum st-hyperflow: a maximum number of hyperedge-disjoint st-hyperpaths.

We show that many of these problems are APX-hard, even in acyclic sequence hypergraphs or with hyperedges of constant length. However, if all the hyperedges are of length at most 2, we show, these problems become polynomially solvable. We also study the special setting in which for every hyperedge there also is a hyperedge with the same sequence, but in the reverse order. Finally, we briefly discuss other algorithmic problems (e.g., finding a minimum spanning tree, or connected components).

Keywords

Colored graphs Labeled problems Oriented hypergraphs Algorithms Complexity 

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Copyright information

© Springer-Verlag GmbH Germany 2016

Authors and Affiliations

  • Kateřina Böhmová
    • 1
  • Jérémie Chalopin
    • 2
  • Matúš Mihalák
    • 1
    • 3
  • Guido Proietti
    • 4
    • 5
  • Peter Widmayer
    • 1
  1. 1.Institute of Theoretical Computer ScienceETH ZürichZürichSwitzerland
  2. 2.LIF, CNRS & Aix-Marseille UniversityMarseilleFrance
  3. 3.Department of Knowledge EngineeringMaastricht UniversityMaastrichtThe Netherlands
  4. 4.DISIM, Università degli Studi dell’AquilaL’AquilaItaly
  5. 5.IASI, CNRRomaItaly

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