The study of the Knot-Free Vertex Deletion problem emerges from its application in the resolution of deadlocks called knots, detected in a classical distributed computation model, that is, the OR-model. A strongly connected subgraph Q of a digraph D with at least two vertices is said to be a knot if there is no arc (u,v) of D with u ∈ V(Q) and v ∉ V(Q) (no-out neighbors of the vertices in Q). Given a directed graph D, the Knot-Free Vertex Deletion (KFVD) problem asks to compute a minimum-size subset S ⊂ V(D) such that D[V⧵S] contains no knots. There is no exact algorithm known for the KFVD problem in the literature that is faster than the trivial O^⋆(2ⁿ) brute-force algorithm. In this paper, we obtain the first non-trivial upper bound for KFVD by designing an exact algorithm running in time 𝒪^⋆(1.576ⁿ), where n is the size of the vertex set in D.
@InProceedings{ramanujan_et_al:LIPIcs.MFCS.2022.78, author = {Ramanujan, M. S. and Sahu, Abhishek and Saurabh, Saket and Verma, Shaily}, title = {{An Exact Algorithm for Knot-Free Vertex Deletion}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {78:1--78:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.78}, URN = {urn:nbn:de:0030-drops-168769}, doi = {10.4230/LIPIcs.MFCS.2022.78}, annote = {Keywords: exact algorithm, knot-free graphs, branching algorithm} }
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