We analyze the computational complexity of optimally playing the two-player board game Push Fight, generalized to an arbitrary board and number of pieces. We prove that the game is PSPACE-hard to decide who will win from a given position, even for simple (almost rectangular) hole-free boards. We also analyze the mate-in-1 problem: can the player win in a single turn? One turn in Push Fight consists of up to two "moves" followed by a mandatory "push". With these rules, or generalizing the number of allowed moves to any constant, we show mate-in-1 can be solved in polynomial time. If, however, the number of moves per turn is part of the input, the problem becomes NP-complete. On the other hand, without any limit on the number of moves per turn, the problem becomes polynomially solvable again.
@InProceedings{bosboom_et_al:LIPIcs.FUN.2018.11, author = {Bosboom, Jeffrey and Demaine, Erik D. and Rudoy, Mikhail}, title = {{Computational Complexity of Generalized Push Fight}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {11:1--11:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-067-5}, ISSN = {1868-8969}, year = {2018}, volume = {100}, editor = {Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.11}, URN = {urn:nbn:de:0030-drops-88029}, doi = {10.4230/LIPIcs.FUN.2018.11}, annote = {Keywords: board games, hardness, mate-in-one} }
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