Integer Concave Cocirculations and Honeycombs

A convex triangular grid is a planar digraph G embedded in the plane so that each bounded face is an equilateral triangle with three edges and their union ${\cal R}$ forms a convex polygon. A function $h:E(G)\to{\mathbb R}$ is called a concave cocirculation if h(e)=g(v)–g(u) for each edge e=(u,v), where g is a concave function on ${\cal R}$ which is affinely linear within each bounded face of G. Knutson and Tao obtained an integrality result on so-called honeycombs implying that if an integer-valued function on the boundary edges is extendable to a concave cocirculation, then it is extendable to an integer one.We show a sharper property: for any concave cocirculation h, there exists an integer concave cocirculation h′ satisfying h′(e)=h(e) for each edge e with $h(e)\in{\mathbb Z}$ contained in the boundary or in a bounded face where h is integer on all edges.Also relevant polyhedral and algorithmic results are presented.