Three counter-examples on semi-graphoids

Authors:
Raymond Hemmecke;Jason Morton;Anne Shiu;Bernd Sturmfels;Oliver Wienand
Affiliations:
Fakultät für mathematik, otto-von-guericke-universität magdeburg, 39106 magdeburg, germany (e-mail: hemmecke@imo.math.uni-magdeburg.de);Department of mathematics, university of california, berkeley, ca 94720, usa (e-mail: mortonj@math.berkeley.edu, annejls@math.berkeley.edu, bernd@math.berkeley.edu);Department of mathematics, university of california, berkeley, ca 94720, usa (e-mail: mortonj@math.berkeley.edu, annejls@math.berkeley.edu, bernd@math.berkeley.edu);Department of mathematics, university of california, berkeley, ca 94720, usa (e-mail: mortonj@math.berkeley.edu, annejls@math.berkeley.edu, bernd@math.berkeley.edu);Fachbereich mathematik, technische universität kaiserslautern, 67653 kaiserslautern, germany (e-mail: wienand@mathematik.uni-kl.de)
Venue:
Combinatorics, Probability and Computing
Year:
2008

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Abstract

Semi-graphoids are combinatorial structures that arise in statistical learning theory. They are equivalent to convex rank tests and to polyhedral fans that coarsen the reflection arrangement of the symmetric group Sn. In this paper we resolve two problems on semi-graphoids posed in Studený's book (2005), and we answer a related question of Postnikov, Reiner and Williams on generalized permutohedra. We also study the semigroup and the toric ideal associated with semi-graphoids.