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Orienteering (with Time Windows) on Restricted Graph Classes

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SOFSEM 2025: Theory and Practice of Computer Science (SOFSEM 2025)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15538))

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Abstract

Given a graph with edge costs and vertex profits and given a budget B, the Orienteering Problem asks for a walk of cost at most B of maximum profit. Additionally, each profit may be given with a time window within which it can be collected by the walk. While the Orienteering Problem and thus the version with time windows are NP-hard in general, it remains open on numerous special graph classes. Since in several applications, especially for planning a route from A to B with waypoints, the input graph can be restricted to tree-like or path-like structures, in this paper we consider orienteering on these graph classes. While the Orienteering Problem with time windows is NP-hard even on undirected paths and cycles, and remains so even if all profits must be collected, we show that for directed paths it can be solved in \(\mathcal {O}(m \log m)\) time (where m is the total number of time windows), even if each profit can be collected in one of several time windows. The same case is shown to be NP-hard for directed cycles.

Particularly interesting is the Orienteering Problem on a directed cycle with one time window per profit. We give an efficient algorithm for the case where all time windows are shorter than the length of the cycle, resulting in a 2-approximation for the general setting. Based on the algorithm for directed paths, we further develop a polynomial-time approximation scheme for this problem. For the case where all profits must be collected, we present an \(\mathcal {O}(n^4)\)-time algorithm. For the Orienteering Problem with time windows for the edges, we give a quadratic time algorithm for undirected paths and observe that the problem is NP-hard for trees.

In the variant without time windows, we show that on trees and thus on graphs with bounded tree-width the Orienteering Problem remains NP-hard. We present, however, an FPT algorithm to solve orienteering with unit profits that we then use to obtain a (\(1+\varepsilon \))-approximation algorithm on graphs with arbitrary profits and bounded tree-width, which improves current results on general graphs.

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Notes

  1. 1.

    \(^*\) The proofs of the results marked with \(*\) are deferred to the full version of this paper [9].

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Authors and Affiliations

  1. Technische Universität Dortmund, Dortmund, Germany

    Kevin Buchin, Mart Hagedoorn, Guangping Li & Carolin Rehs

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  1. Kevin Buchin
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Correspondence to Mart Hagedoorn .

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  1. Comenius University in Bratislava, Bratislava, Slovakia

    Rastislav Královič

  2. Czech Academy of Sciences, Prague, Czech Republic

    Věra Kůrková

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Buchin, K., Hagedoorn, M., Li, G., Rehs, C. (2025). Orienteering (with Time Windows) on Restricted Graph Classes. In: Královič, R., Kůrková, V. (eds) SOFSEM 2025: Theory and Practice of Computer Science. SOFSEM 2025. Lecture Notes in Computer Science, vol 15538. Springer, Cham. https://doi.org/10.1007/978-3-031-82670-2_12

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