# Multistage s-t Path: Confronting Similarity with Dissimilarity

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 resulting 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 the two variants of Multistage s-t Path are already NP-hard for an input sequence of only two graphs. 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, we prove parameterized hardness (W[1]-hardness) regarding the size of the path sequence (solution size) for both variants, vertex- and edge-similarity. As a novel conceptual contribution, we then modify the multistage model by asking for dissimilar consecutive paths. As one of the main results, we prove that dissimilarity allows for fixed-parameter tractability for the parameter solution size, thereby contrasting our W[1]-hardness proof of the corresponding similarity case.

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