Multistage s–t Path: Confronting Similarity with Dissimilarity

Abstract Addressing a quest by Gupta et al. (in: Proceedings of the 41st international colloquium on automata, languages, and programming (ICALP 2014), vol 8572 LNCS. Springer, pp 563–575, we provide first, comprehensive study finding short s – t path in multistage graph model, referred to as Multistage Path problem. Herein, given sequence graphs over same vertex set but changing edge sets, task is find paths each (“snapshot”) such that found consecutive are “similar”. We measure similarity size symmetric difference either (vertex-similarity) or (edge-similarity) any two paths. prove these variants already $${\text {NP}}$$ NP -hard for an input only snapshots maximum degree four. Motivated this fact natural applications scenario e.g. traffic route planning, perform parameterized complexity analysis. Among other results, both variants, vertex- edge-similarity, hardness ( {W[1]}}$$ W[1] -hardness) regarding parameter length (solution size). As further conceptual investigation, then modify model asking dissimilar one main technical results (employing so-called representative sets known from non-temporal settings), dissimilarity allows fixed-parameter tractability solution size, contrasting with our W[1]-hardness proof corresponding case. also partially positive concerning efficient effective data reduction (kernelization).