Concave cocirculations in a triangular grid

Abstract

Let $G=(V(G),E(G))$ be a planar digraph embedded in the plane in which all inner faces are equilateral triangles (with three edges in each), and let the union $\Rscr$ of these faces forms a convex polygon. The question is: given a function $\sigma$ on the boundary edges of $G$, does there exist a concave function $f$ on $\Rscr$ which is affinely linear within each bounded face and satisfies $f(v)-f(u)=\sigma(e)$ for each boundary edge $e=(u,v)$? The functions $\sigma$ admitting such an $f$ form a polyhedral cone $C$, and when the region $\Rscr$ is a triangle, $C$ turns out to be exactly the cone of boundary data of honeycombs. Studing honeycombs in connection with a problem on spectra of triples of zero-sum Hermitian matrices, Knutson, Tao, and Woodward \cite{KTW} showed that $C$ is described by linear inequalities of Horn's type with respect to so-called {\em puzzles}, along with obvious linear constraints. The purpose of this paper is to give an alternative proof of that result, working in terms of discrete concave finctions, rather than honeycombs, and using only linear programming and combinatorial tools. Moreover, we extend the result to an arbitrary convex polygon $\Rscr$.