Sequence Hypergraphs: Paths, Flows, and Cuts

Kateřina Böhmová, Jérémie Chalopin, Matúš Mihalák, Guido Proietti, Peter Widmayer

Research output: Chapter in Book/Report/Conference proceedingChapterAcademic

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 (not unlike a directed path). Sequence hypergraphs are motivated by problems in public transportation networks, as they conveniently represent transportation lines. We study the complexity of several fundamental 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 that 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 reverse order. Finally, we briefly discuss other algorithmic problems such as finding a minimum spanning tree, or connected components.
Original languageEnglish
Title of host publicationAdventures Between Lower Bounds and Higher Altitudes
EditorsH. J. Böckenhauer, D. Komm, W. Unger
PublisherSpringer
ChapterChapter 12
Pages191-215
DOIs
Publication statusPublished - 9 Aug 2018

Publication series

SeriesLecture Notes in Computer Science
Volume11011
ISSN0302-9743

Cite this

Böhmová, K., Chalopin, J., Mihalák, M., Proietti, G., & Widmayer, P. (2018). Sequence Hypergraphs: Paths, Flows, and Cuts. In H. J. Böckenhauer, D. Komm, & W. Unger (Eds.), Adventures Between Lower Bounds and Higher Altitudes (pp. 191-215). Springer. Lecture Notes in Computer Science, Vol.. 11011 https://doi.org/10.1007/978-3-319-98355-4_12