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 R forms a convex polygon. A function h : E(G) → 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 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) ∈ 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.