TY - JOUR
T1 - Bipartite perfect matching as a real polynomial
AU - Beniamini, Gal
AU - Nisan, Noam
N1 - Publisher Copyright:
© 2023, The Hebrew University of Jerusalem.
PY - 2023/9
Y1 - 2023/9
N2 - We obtain a description of the Bipartite Perfect Matching decision problem as a multilinear polynomial over the Reals. We show that it has full total degree and (1−o(1))⋅2n2 monomials with non-zero coefficients. In contrast, we show that in the dual representation (switching the roles of 0 and 1) the number of monomials is only exponential in Θ(n log n). Our proof relies heavily on the fact that the lattice of graphs which are “matching-covered” is Eulerian.
AB - We obtain a description of the Bipartite Perfect Matching decision problem as a multilinear polynomial over the Reals. We show that it has full total degree and (1−o(1))⋅2n2 monomials with non-zero coefficients. In contrast, we show that in the dual representation (switching the roles of 0 and 1) the number of monomials is only exponential in Θ(n log n). Our proof relies heavily on the fact that the lattice of graphs which are “matching-covered” is Eulerian.
UR - http://www.scopus.com/inward/record.url?scp=85173696994&partnerID=8YFLogxK
U2 - 10.1007/s11856-023-2505-9
DO - 10.1007/s11856-023-2505-9
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AN - SCOPUS:85173696994
SN - 0021-2172
VL - 256
SP - 91
EP - 131
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -