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).
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Notes
- 1.
To avoid confusion with directed hypergraphs, we prefer the term sequence hypergraphs to refer to the hypergraphs with hyperedges formed as sequences of vertices.
- 2.
Note, if \(E'\) is a backward hyperedge of E, also E is a backward hyperedge of \(E'\).
- 3.
An st-hyperpath \(P^*\) and its underlying path are defined as in sequence hypergraphs.
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Acknowledgements
Kateřina Böhmová is supported by a Google Europe Fellowship in Optimization Algorithms. Jérémie Chalopin was partially supported by the ANR project MACARON (anr-13-js02-0002).
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Böhmová, K., Chalopin, J., Mihalák, M., Proietti, G., Widmayer, P. (2016). Sequence Hypergraphs. In: Heggernes, P. (eds) Graph-Theoretic Concepts in Computer Science. WG 2016. Lecture Notes in Computer Science(), vol 9941. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53536-3_24
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DOI: https://doi.org/10.1007/978-3-662-53536-3_24
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