88 research outputs found
Discrete convexity and unimodularity. I
In this paper we develop a theory of convexity for a free Abelian group M
(the lattice of integer points), which we call theory of discrete convexity. We
characterize those subsets X of the group M that could be call "convex". One
property seems indisputable: X should coincide with the set of all integer
points of its convex hull co(X) (in the ambient vector space V). However, this
is a first approximation to a proper discrete convexity, because such
non-intersecting sets need not be separated by a hyperplane. This issue is
closely related to the question when the intersection of two integer polyhedra
is an integer polyhedron. We show that unimodular systems (or more generally,
pure systems) are in one-to-one correspondence with the classes of discrete
convexity. For example, the well-known class of g-polymatroids corresponds to
the class of discrete convexity associated to the unimodular system A_n:={\pm
e_i, e_i-ej} in Z^n.Comment: 26 pages, Late
Condorcet domains of tiling type
A Condorcet domain (CD) is a collection of linear orders on a set of
candidates satisfying the following property: for any choice of preferences of
voters from this collection, a simple majority rule does not yield cycles. We
propose a method of constructing "large" CDs by use of rhombus tiling diagrams
and explain that this method unifies several constructions of CDs known
earlier. Finally, we show that three conjectures on the maximal sizes of those
CDs are, in fact, equivalent and provide a counterexample to them.Comment: 16 pages. To appear in Discrete Applied Mathematic
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