241 research outputs found
Parity of transversals of Latin squares
We introduce a notion of parity for transversals, and use it to show that in
Latin squares of order , the number of transversals is a multiple of
4. We also demonstrate a number of relationships (mostly congruences modulo 4)
involving , where is the number of diagonals of a given
Latin square that contain exactly different symbols.
Let denote the matrix obtained by deleting row and column
from a parent matrix . Define to be the number of transversals
in , for some fixed Latin square . We show that for all and . Also, if has odd order then the
number of transversals of equals mod 2. We conjecture that for all .
In the course of our investigations we prove several results that could be of
interest in other contexts. For example, we show that the number of perfect
matchings in a -regular bipartite graph on vertices is divisible by
when is odd and . We also show that for all , when is an integer matrix of odd
order with all row and columns sums equal to
A module-theoretic approach to matroids
Speyer recognized that matroids encode the same data as a special class of
tropical linear spaces and Shaw interpreted tropically certain basic matroid
constructions; additionally, Frenk developed the perspective of tropical linear
spaces as modules over an idempotent semifield. All together, this provides
bridges between the combinatorics of matroids, the algebra of idempotent
modules, and the geometry of tropical linear spaces. The goal of this paper is
to strengthen and expand these bridges by systematically developing the
idempotent module theory of matroids. Applications include a geometric
interpretation of strong matroid maps and the factorization theorem; a
generalized notion of strong matroid maps, via an embedding of the category of
matroids into a category of module homomorphisms; a monotonicity property for
the stable sum and stable intersection of tropical linear spaces; a novel
perspective of fundamental transversal matroids; and a tropical analogue of
reduced row echelon form.Comment: 22 pages; v3 minor corrections/clarifications; to appear in JPA
Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces
This paper transfers a randomized algorithm, originally used in geometric
optimization, to computational problems in commutative algebra. We show that
Clarkson's sampling algorithm can be applied to two problems in computational
algebra: solving large-scale polynomial systems and finding small generating
sets of graded ideals. The cornerstone of our work is showing that the theory
of violator spaces of G\"artner et al.\ applies to polynomial ideal problems.
To show this, one utilizes a Helly-type result for algebraic varieties. The
resulting algorithms have expected runtime linear in the number of input
polynomials, making the ideas interesting for handling systems with very large
numbers of polynomials, but whose rank in the vector space of polynomials is
small (e.g., when the number of variables and degree is constant).Comment: Minor edits, added two references; results unchange
Computing the vertices of tropical polyhedra using directed hypergraphs
We establish a characterization of the vertices of a tropical polyhedron
defined as the intersection of finitely many half-spaces. We show that a point
is a vertex if, and only if, a directed hypergraph, constructed from the
subdifferentials of the active constraints at this point, admits a unique
strongly connected component that is maximal with respect to the reachability
relation (all the other strongly connected components have access to it). This
property can be checked in almost linear-time. This allows us to develop a
tropical analogue of the classical double description method, which computes a
minimal internal representation (in terms of vertices) of a polyhedron defined
externally (by half-spaces or hyperplanes). We provide theoretical worst case
complexity bounds and report extensive experimental tests performed using the
library TPLib, showing that this method outperforms the other existing
approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section
5 (using directed hypergraphs), detailed appendix; v3: major revision of the
article (adding tropical hyperplanes, alternative method by arrangements,
etc); v4: minor revisio
- …