Sequence Hypergraphs: Paths, Flows, and Cuts

Kateřina Böhmová (1) , Jérémie Chalopin (2) , Matúš Mihalák (3) , Guido Proietti (4, 5) , Peter Widmayer (1)

1 Institute of Theoretical Computer Science [Zurich]
2 DALGO - Algorithmique Distribuée
3 Department of data science and Knowledge Engineering [Maastricht]
4 DISIM - Department of Information Engineering, Computer Science and Mathematics = Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica [L'Aquila]
5 IASI - Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti”

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.

Mots clés: Sequence hypergraphs; colored graphs; labeled problems; transportation lines; algorithms; complexity

Domaines: Mathématique discrète [cs.DM]; Algorithme et structure de données [cs.DS]

Kateřina Böhmová, Jérémie Chalopin, Matúš Mihalák, Guido Proietti, Peter Widmayer. Sequence Hypergraphs: Paths, Flows, and Cuts. Hans-Joachim Böckenhauer; Dennis Komm; Walter Unger. Adventures Between Lower Bounds and Higher Altitudes, 11011, Springer, pp.191-215, 2018, 978-3-319-98355-4. ⟨10.1007/978-3-319-98355-4_12⟩. ⟨hal-02065856⟩