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Planar flows and quadratic relations over semirings | | Vladimir I. Danilov
; Alexander V. Karzanov
; Gleb A. Koshevoy
; | | Date: |
13 Feb 2011 | | Abstract: | It is well known, due to Lindstr"om, that the minors of a matrix can be
expressed in terms of weights of flows in a planar directed graph. Another fact
is that there are plenty of quadratic relations involving minors, of which
Pl"ucker ones are most popular. Generalizing and unifying these facts and
their tropical counterparts, we consider a wide class of functions which are
generated by flows in a planar graph and take values in an arbitrary
commutative semiring. We show that the "universal" quadratic relations
performed by such functions can be described in terms of certain matchings, and
as a consequence, give combinatorial necessary and sufficient conditions for
these relations. Also we give applications and discuss related topics. | | Source: | arXiv, 1102.2578 | | Services: | Forum | Review | PDF | Favorites | |
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