Skip to main content Accessibility help
×
Home
Hostname: page-component-78bd46657c-2pqp7 Total loading time: 0.363 Render date: 2021-05-08T10:31:03.708Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Three Counter-Examples on Semi-Graphoids

Published online by Cambridge University Press:  01 March 2008

RAYMOND HEMMECKE
Affiliation:
Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg, 39106 Magdeburg, Germany (e-mail: hemmecke@imo.math.uni-magdeburg.de)
JASON MORTON
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA (e-mail: mortonj@math.berkeley.edu, annejls@math.berkeley.edu, bernd@math.berkeley.edu)
ANNE SHIU
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA (e-mail: mortonj@math.berkeley.edu, annejls@math.berkeley.edu, bernd@math.berkeley.edu)
BERND STURMFELS
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA (e-mail: mortonj@math.berkeley.edu, annejls@math.berkeley.edu, bernd@math.berkeley.edu)
OLIVER WIENAND
Affiliation:
Fachbereich Mathematik, Technische Universität Kaiserslautern, 67653 Kaiserslautern, Germany (e-mail: wienand@mathematik.uni-kl.de)

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.

Type
Paper
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below.

References

[1] 4ti2: A software package for algebraic, geometric and combinatorial problems on linear spaces. Available at: http://www.4ti2.de.Google Scholar
[2]Aoki, S., Takemura, A. and Yoshida, R. (2005) Indispensable monomials of toric ideals and Markov bases. In Proc Asian Symposium on Computer Mathematics: ASCM 2005 (S. Pae and H. Park, eds), Korea Institute for Advanced Study, pp. 200–202Google Scholar
[3]Bokowski, J. and Sturmfels, B. (1987) Polytopal and non-polytopal spheres: An algorithmic approach. Israel J. Math. 57 257271.CrossRefGoogle Scholar
[4]Bruns, W. and Koch, R. (2001) Computing the integral closure of an affine semigroup. In Effective Methods in Algebraic and Analytic Geometry: Kraków 2000. Univ. Iagel. Acta Math. 39 5970.Google Scholar
[5]Diaconis, P. and Sturmfels, B. (1998) Algebraic algorithms for sampling from conditional distributions. Ann. Statist. 26 363397.Google Scholar
[6]Eisenbud, D. (1995) Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, Springer, New York.Google Scholar
[7]Gawrilow, E. and Joswig, M. (2000) Polymake: A framework for analyzing convex polytopes. In Polytopes: Combinatorics and Computation (Kalai, G. and Ziegler, G. M., eds), Birkhäuser, pp. 4374.CrossRefGoogle Scholar
[8]Geiger, D., Meek, C. and Sturmfels, B. (2006) On the toric algebra of graphical models. Ann. Statist. 34 14631492.CrossRefGoogle Scholar
[9]Grayson, D. and Stillman, M. Macaulay2: A software system for research in algebraic geometry. Available at: http://www.math.uiuc.edu/Macaulay2/.Google Scholar
[10]Hemmecke, R., Takemura, A. and Yoshida, R. Computing holes in semi-groups. Preprint available at: math.CO/0607599.Google Scholar
[11]Hirai, H. (2006) Sequences of stellar subdivisions. Preprint.Google Scholar
[12]Matúš, F. (1999) Conditional independences among four random variables III: Final conclusion. Combin. Probab. Comput. 8 269276.CrossRefGoogle Scholar
[13]Matúš, F. (2003) Conditional probabilities and permutohedron. Ann. Inst. H. Poincaré Probab. Statist. 39 687701.CrossRefGoogle Scholar
[14]Matúš, F. (2004) Towards classification of semi-graphoids. Discrete Math. 277 115145.CrossRefGoogle Scholar
[15]Miller, E. and Sturmfels, B. (2004) Combinatorial Commutative Algebra, Graduate Texts in Mathematics, Springer, New York.Google Scholar
[16]Morton, J., Pachter, L., Shiu, A., Sturmfels, B. and Wienand, O. (2006) Geometry of rank tests. Probabilistic Graphical Models (PGM 3), Prague 2006. Available at: math.ST/0605173.Google Scholar
[17]Postnikov, A. (2005) Permutohedra, associahedra, and beyond. Preprint available at: math.CO/0507163.Google Scholar
[18]Postnikov, A., Reiner, V. and Williams, L. Faces of simple generalized permutohedra. Preprint available at: math.CO/0609184.Google Scholar
[19]Studený, M. (1994) Structural semigraphoids. Internat. J. General Systems 22 207217.CrossRefGoogle Scholar
[20]Studený, M. (2005) Probabilistic Conditional Independence Structures, Springer Series in Information Science and Statistics, Springer, London.Google Scholar
[21]Sturmfels, B. (1996) Gröbner Bases and Convex Polytopes, AMS, Providence.Google Scholar
[22]Ziegler, G. (1998) Lectures on Polytopes. Graduate Texts in Mathematics, Springer, New York.Google Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Three Counter-Examples on Semi-Graphoids
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Three Counter-Examples on Semi-Graphoids
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Three Counter-Examples on Semi-Graphoids
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *