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Integer Concave Cocirculations and Honeycombs

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Part of the Lecture Notes in Computer Science book series (LNCS,volume 3064)

Abstract

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.

Keywords

  • Planar graph
  • Discrete convex (concave) function
  • Honeycomb

AMS Subject Classification

  • 90C10
  • 90C27
  • 05C99

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References

  1. Buch, A.S.: The saturation conjecture (after A. Knutson and T. Tao). With an appendix by William Fulton. Enseign. Math (2) 46(1-2), 43–60 (2000)

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  3. Karzanov, A.V.: Concave cocirculations in a triangular grid, submitted to Linear Algebra and Appl.

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  4. Knutson, A., Tao, T.: The honeycomb model of GLn(C) tensor products I: Proof of the saturation conjecture. J. Amer. Math. Soc. 12(4), 1055–1090 (1999)

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  5. Knutson, A., Tao, T., Woodward, C.: The honeycomb model of GLKnutson, A., Tao, T., Woodward, C.: The honeycomb model of GL n (ℂ) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone, Preprint, 2001; to appear in J. Amer. Math. Soc. (2001)

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© 2004 Springer-Verlag Berlin Heidelberg

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Karzanov, A.V. (2004). Integer Concave Cocirculations and Honeycombs. In: Bienstock, D., Nemhauser, G. (eds) Integer Programming and Combinatorial Optimization. IPCO 2004. Lecture Notes in Computer Science, vol 3064. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-25960-2_28

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  • DOI: https://doi.org/10.1007/978-3-540-25960-2_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22113-5

  • Online ISBN: 978-3-540-25960-2

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