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Problemy Peredachi Informatsii, 2021, Volume 57, Issue 2, Pages 51–70
DOI: https://doi.org/10.31857/S0555292321020042 (Mi ppi2341) 		 
Large Systems

Counting the number of perfect matchings, and generalized decision trees

M. N. Vyalyi

National Research University—Higher School of Economics, Moscow, Russia
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DOI: https://doi.org/10.31857/S0555292321020042
Abstract: We consider a generalization of the Pólya–Kasteleyn approach to counting the number of perfect matchings in a graph based on computing the symbolic Pfaffian of a directed adjacency matrix of the graph. Complexity of algorithms based on this approach is related to the complexity of the sign function of a perfect matching in generalized decision tree models. We obtain lower bounds on the complexity of the sign of a perfect matching in terms of the deterministic communication complexity of the XOR sign function of a matching. These bounds demonstrate limitations of the symbolic Pfaffian method for both the general case and the case of sparse graphs.
Keywords: perfect matching, Pfaffian, decision tree, communication complexity.
Funding agency Grant number
Russian Science Foundation 20-11-20203
The research was carried out at the expense of the Russian Science Foundation, project no. 20-11-20203.
Received: 19.09.2020
Revised: 17.12.2020
Accepted: 17.12.2020
English version:
Problems of Information Transmission, 2021, Volume 57, Issue 2, Pages 143–160
DOI: https://doi.org/10.1134/S0032946021020046
Bibliographic databases:
Document Type: Article
UDC: 621.391 : 519.178
Language: Russian
Citation: M. N. Vyalyi, “Counting the number of perfect matchings, and generalized decision trees”, Probl. Peredachi Inf., 57:2 (2021), 51–70; Problems Inform. Transmission, 57:2 (2021), 143–160
Citation in format AMSBIB
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\pages 51--70
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    		Проблемы передачи информации Problems of Information Transmission
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