## Abstract

Given a graph (Formula presented.), a k-Path Cover is defined as a subset C of the nodes V such that every simple path in G consisting of k nodes contains at least one node from C. Similarly, a k-Shortest Path Cover has to contain at least one node of every shortest path in G that consists of k nodes. In this paper, we extend the notion of k-(Shortest) Path Covers such that the objects to be covered don't have to be single paths but can be concatenations of up to p simple (or shortest) paths. For the generalized problem of computing concatenated (Formula presented.) -Shortest Path Covers, we present theoretical results regarding the VC-dimension of the concatenated path set in dependency of p in undirected as well as directed graphs. By proving a low VC-dimension in both settings, we enable the design of efficient approximation algorithms. Furthermore, we discuss how a pruning algorithm originally developed for k-Path Cover computation can be abstracted and modified in order to also solve concatenated (Formula presented.) -Path Cover problems. A crucial ingredient for the pruning algorithm to work efficiently is a path concatenation recognition algorithm. We describe general recognition algorithms for simple path concatenations as well as shortest path concatenations. Subsequently, we present more refined results for interesting special cases as piecewise shortest paths, hyperpaths, round tours, and trees. An extensive experimental study on different graph types proves the applicability and efficiency of our approaches.

Original language | English |
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Pages (from-to) | 32-56 |

Number of pages | 25 |

Journal | International Journal of Computer Mathematics: Computer Systems Theory |

Volume | 8 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2 Jan 2023 |

Event | 27th International Computing and Combinatorics Conference, COCOON 2021 - http://cocoon-conference.org/2021/, Tainan, Taiwan Duration: 24 Oct 2021 → 26 Oct 2021 |

## Scopus Subject Areas

- Computational Mathematics
- Computational Theory and Mathematics

## User-Defined Keywords

- hitting set
- Shortest paths
- vc-dimension