Algebraic Representations of Unique Bipartite Perfect Matching
We obtain complete characterizations of the Unique Bipartite Perfect Matching function, and of its Boolean dual, using multilinear polynomials over the reals. Building on previous results [Beniamini, 2020; Beniamini and Nisan, 2021], we show that, surprisingly, the dual description is sparse and has low đâ-norm - only exponential in Î(n log n), and this result extends even to other families of matching-related functions. Our approach relies on the MĂ¶bius numbers in the matching-covered lattice, and a key ingredient in our proof is MĂ¶bius' inversion formula.
These polynomial representations yield complexity-theoretic results. For instance, we show that unique bipartite matching is evasive for classical decision trees, and nearly evasive even for generalized query models. We also obtain a tight Î(n log n) bound on the log-rank of the associated two-party communication task.
Bipartite Perfect Matching
Boolean Functions
Partially Ordered Sets
Mathematics of computing~Matchings and factors
Theory of computation~Communication complexity
Theory of computation~Oracles and decision trees
16:1-16:17
Regular Paper
Gal
Beniamini
Gal Beniamini
The Hebrew University of Jerusalem, Israel
10.4230/LIPIcs.MFCS.2022.16
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Gal Beniamini
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