An st-shortest path, or st-path for short, in a graph G is a shortest (induced) path from s to t in G. Two st-paths are said to be adjacent if they differ on exactly one vertex. A reconfiguration sequence between two st-paths P and Q is a sequence of adjacent st-paths starting from P and ending at Q. Deciding whether there exists a reconfiguration sequence between two given st-paths is known to be PSPACE-complete, even on restricted classes of graphs such as graphs of bounded bandwidth (hence pathwidth). On the positive side, and rather surprisingly, the problem is polynomial-time solvable on planar graphs. In this paper, we study the parameterized complexity of the Shortest Path Reconfiguration (SPR) problem. We show that SPR is W[1]-hard parameterized by k + 𝓁, even when restricted to graphs of bounded (constant) degeneracy; here k denotes the number of edges on an st-path, and 𝓁 denotes the length of a reconfiguration sequence from P to Q. We complement our hardness result by establishing the fixed-parameter tractability of SPR parameterized by 𝓁 and restricted to nowhere-dense classes of graphs. Additionally, we establish fixed-parameter tractability of SPR when parameterized by the treedepth, by the cluster-deletion number, or by the modular-width of the input graph.
@InProceedings{bousquet_et_al:LIPIcs.IPEC.2024.23, author = {Bousquet, Nicolas and Gajjar, Kshitij and Lahiri, Abhiruk and Mouawad, Amer E.}, title = {{Parameterized Shortest Path Reconfiguration}}, booktitle = {19th International Symposium on Parameterized and Exact Computation (IPEC 2024)}, pages = {23:1--23:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-353-9}, ISSN = {1868-8969}, year = {2024}, volume = {321}, editor = {Bonnet, \'{E}douard and Rz\k{a}\.{z}ewski, Pawe{\l}}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2024.23}, URN = {urn:nbn:de:0030-drops-222491}, doi = {10.4230/LIPIcs.IPEC.2024.23}, annote = {Keywords: combinatorial reconfiguration, shortest path reconfiguration, parameterized complexity, structural parameters, treedepth, cluster deletion number, modular width} }
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