The exact complexity of solving parity games is a major open problem. Several authors have searched for efficient algorithms over specific classes of graphs. In particular, Obdržálek showed that for graphs of bounded tree-width or clique-width, the problem is in P, which was later improved by Ganardi, who showed that it is even in LOGCFL (with an additional assumption for clique-width case). Here we extend this line of research by showing that for graphs of bounded tree-depth the problem of solving parity games is in logspace uniform AC⁰. We achieve this by first considering a parameter that we obtain from a modification of clique-width, which we call shallow clique-width. We subsequently provide a suitable reduction.
@InProceedings{staniszewski:LIPIcs.CSL.2023.33, author = {Staniszewski, Konrad}, title = {{Parity Games of Bounded Tree-Depth}}, booktitle = {31st EACSL Annual Conference on Computer Science Logic (CSL 2023)}, pages = {33:1--33:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-264-8}, ISSN = {1868-8969}, year = {2023}, volume = {252}, editor = {Klin, Bartek and Pimentel, Elaine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.33}, URN = {urn:nbn:de:0030-drops-174942}, doi = {10.4230/LIPIcs.CSL.2023.33}, annote = {Keywords: Parity Games, Circuits, Tree-Depth, Clique-Width} }
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