Integer Concave Cocirculations and Honeycombs

  • Alexander V. Karzanov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3064)


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.


Planar graph Discrete convex (concave) function Honeycomb 

AMS Subject Classification

90C10 90C27 05C99 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Alexander V. Karzanov
    • 1
  1. 1.Institute for System AnalysisMoscowRussia

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