Multistage st Path: Confronting Similarity with Dissimilarity
Abstract
Addressing a quest by Gupta et al. [ICALP'14], we provide a first, comprehensive study of finding a short st path in the multistage graph model, referred to as the Multistage st Path problem. Herein, given a sequence of graphs over the same vertex set but changing edge sets, the task is to find short st paths in each graph ("snapshot") such that in the found path sequence the consecutive st paths are "similar". We measure similarity by the size of the symmetric difference of either the vertex set (vertexsimilarity) or the edge set (edgesimilarity) of any two consecutive paths. We prove that these two variants of Multistage st Path are already NPhard 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 edgesimilarity, we prove parameterized hardness (W[1]hardness) regarding the parameter path length (solution size) for both variants, vertex and edgesimilarity. As a further conceptual study, we then modify the multistage model by asking for dissimilar consecutive paths. As one of the main technical results (employing socalled representative sets known from nontemporal settings), we prove that dissimilarity allows for fixedparameter tractability for the parameter solution size, contrasting our W[1]hardness proof of the corresponding similarity case. We also provide partially positive results concerning efficient and effective data reduction (kernelization).
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 arXiv:
 arXiv:2002.07569
 Bibcode:
 2020arXiv200207569F
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms;
 68R10;
 68Q17;
 68Q25;
 68W40