For vertices u and v of an n-vertex graph G, a uv-trail of G is an induced uv-path of G that is not a shortest uv-path of G. Berger, Seymour, and Spirkl [Discrete Mathematics 2021] gave the previously only known polynomial-time algorithm, running in O(n^{18}) time, to either output a uv-trail of G or ensure that G admits no uv-trail. We reduce the complexity to the time required to perform a poly-logarithmic number of multiplications of n²× n² Boolean matrices, leading to a largely improved O(n^{4.75})-time algorithm.
@InProceedings{chiu_et_al:LIPIcs.STACS.2022.23, author = {Chiu, Yung-Chung and Lu, Hsueh-I}, title = {{Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-Shortest Induced Paths}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {23:1--23:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.23}, URN = {urn:nbn:de:0030-drops-158333}, doi = {10.4230/LIPIcs.STACS.2022.23}, annote = {Keywords: Induced subgraph, induced path, non-shortest path, dynamic data structure} }
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