eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-12-04
43:1
43:16
10.4230/LIPIcs.ISAAC.2020.43
article
Multistage s-t Path: Confronting Similarity with Dissimilarity in Temporal Graphs
Fluschnik, Till
1
https://orcid.org/0000-0003-2203-4386
Niedermeier, Rolf
1
https://orcid.org/0000-0003-1703-1236
Schubert, Carsten
1
Zschoche, Philipp
1
https://orcid.org/0000-0001-9846-0600
Technische Universität Berlin, Algorithmics and Computational Complexity, Germany
Addressing a quest by Gupta et al. [ICALP'14], we provide a first, comprehensive study of finding a short s-t path in the multistage graph model, referred to as the Multistage s-t Path problem. Herein, given a sequence of graphs over the same vertex set but changing edge sets, the task is to find short s-t paths in each graph ("snapshot") such that in the found path sequence the consecutive s-t paths are "similar". We measure similarity by the size of the symmetric difference of either the vertex set (vertex-similarity) or the edge set (edge-similarity) of any two consecutive paths. We prove that these two variants of Multistage s-t Path are already NP-hard for an input sequence of only two graphs and maximum vertex degree four. Motivated by this fact and natural applications of this scenario e.g. in traffic route planning, we perform a parameterized complexity analysis. Among other results, for both variants, vertex- and edge-similarity, we prove parameterized hardness (W[1]-hardness) regarding the parameter path length (solution size) for both variants, vertex- and edge-similarity. As a further conceptual study, we then modify the multistage model by asking for dissimilar consecutive paths. One of our main technical results (employing so-called representative sets known from non-temporal settings) is that dissimilarity allows for fixed-parameter tractability for the parameter solution size, contrasting the W[1]-hardness of the corresponding similarity case. We also provide partially positive results concerning efficient and effective data reduction (kernelization).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol181-isaac2020/LIPIcs.ISAAC.2020.43/LIPIcs.ISAAC.2020.43.pdf
Temporal graphs
shortest paths
consecutive similarity
consecutive dissimilarity
parameterized complexity
kernelization
representative sets in temporal graphs