---
_id: '1092'
abstract:
- lang: eng
  text: 'A graphical model encodes conditional independence relations via the Markov
    properties. For an undirected graph these conditional independence relations can
    be represented by a simple polytope known as the graph associahedron, which can
    be constructed as a Minkowski sum of standard simplices. We show that there is
    an analogous polytope for conditional independence relations coming from a regular
    Gaussian model, and it can be defined using multiinformation or relative entropy.
    For directed acyclic graphical models we give a construction of this polytope
    as a Minkowski sum of matroid polytopes. Finally, we apply this geometric insight
    to construct a new ordering-based search algorithm for causal inference via directed
    acyclic graphical models. '
author:
- first_name: Fatemeh
  full_name: Mohammadi, Fatemeh
  id: 2C29581E-F248-11E8-B48F-1D18A9856A87
  last_name: Mohammadi
- first_name: Caroline
  full_name: Uhler, Caroline
  id: 49ADD78E-F248-11E8-B48F-1D18A9856A87
  last_name: Uhler
  orcid: 0000-0002-7008-0216
- first_name: Charles
  full_name: Wang, Charles
  last_name: Wang
- first_name: Josephine
  full_name: Yu, Josephine
  last_name: Yu
citation:
  ama: Mohammadi F, Uhler C, Wang C, Yu J. Generalized permutohedra from probabilistic
    graphical models. <i>SIAM Journal on Discrete Mathematics</i>. 2018;32(1):64-93.
    doi:<a href="https://doi.org/10.1137/16M107894X">10.1137/16M107894X</a>
  apa: Mohammadi, F., Uhler, C., Wang, C., &#38; Yu, J. (2018). Generalized permutohedra
    from probabilistic graphical models. <i>SIAM Journal on Discrete Mathematics</i>.
    SIAM. <a href="https://doi.org/10.1137/16M107894X">https://doi.org/10.1137/16M107894X</a>
  chicago: Mohammadi, Fatemeh, Caroline Uhler, Charles Wang, and Josephine Yu. “Generalized
    Permutohedra from Probabilistic Graphical Models.” <i>SIAM Journal on Discrete
    Mathematics</i>. SIAM, 2018. <a href="https://doi.org/10.1137/16M107894X">https://doi.org/10.1137/16M107894X</a>.
  ieee: F. Mohammadi, C. Uhler, C. Wang, and J. Yu, “Generalized permutohedra from
    probabilistic graphical models,” <i>SIAM Journal on Discrete Mathematics</i>,
    vol. 32, no. 1. SIAM, pp. 64–93, 2018.
  ista: Mohammadi F, Uhler C, Wang C, Yu J. 2018. Generalized permutohedra from probabilistic
    graphical models. SIAM Journal on Discrete Mathematics. 32(1), 64–93.
  mla: Mohammadi, Fatemeh, et al. “Generalized Permutohedra from Probabilistic Graphical
    Models.” <i>SIAM Journal on Discrete Mathematics</i>, vol. 32, no. 1, SIAM, 2018,
    pp. 64–93, doi:<a href="https://doi.org/10.1137/16M107894X">10.1137/16M107894X</a>.
  short: F. Mohammadi, C. Uhler, C. Wang, J. Yu, SIAM Journal on Discrete Mathematics
    32 (2018) 64–93.
date_created: 2018-12-11T11:50:06Z
date_published: 2018-01-01T00:00:00Z
date_updated: 2021-01-12T06:48:13Z
day: '01'
doi: 10.1137/16M107894X
extern: '1'
intvolume: '        32'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1606.01814
month: '01'
oa: 1
oa_version: Preprint
page: 64-93
publication: SIAM Journal on Discrete Mathematics
publication_status: published
publisher: SIAM
publist_id: '6284'
quality_controlled: '1'
status: public
title: Generalized permutohedra from probabilistic graphical models
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 32
year: '2018'
...
---
_id: '1547'
abstract:
- lang: eng
  text: Let G be a graph on the vertex set V(G) = {x1,…,xn} with the edge set E(G),
    and let R = K[x1,…, xn] be the polynomial ring over a field K. Two monomial ideals
    are associated to G, the edge ideal I(G) generated by all monomials xixj with
    {xi,xj} ∈ E(G), and the vertex cover ideal IG generated by monomials ∏xi∈Cxi for
    all minimal vertex covers C of G. A minimal vertex cover of G is a subset C ⊂
    V(G) such that each edge has at least one vertex in C and no proper subset of
    C has the same property. Indeed, the vertex cover ideal of G is the Alexander
    dual of the edge ideal of G. In this paper, for an unmixed bipartite graph G we
    consider the lattice of vertex covers LG and we explicitly describe the minimal
    free resolution of the ideal associated to LG which is exactly the vertex cover
    ideal of G. Then we compute depth, projective dimension, regularity and extremal
    Betti numbers of R/I(G) in terms of the associated lattice.
author:
- first_name: Fatemeh
  full_name: Mohammadi, Fatemeh
  id: 2C29581E-F248-11E8-B48F-1D18A9856A87
  last_name: Mohammadi
- first_name: Somayeh
  full_name: Moradi, Somayeh
  last_name: Moradi
citation:
  ama: Mohammadi F, Moradi S. Resolution of unmixed bipartite graphs. <i>Bulletin
    of the Korean Mathematical Society</i>. 2015;52(3):977-986. doi:<a href="https://doi.org/10.4134/BKMS.2015.52.3.977">10.4134/BKMS.2015.52.3.977</a>
  apa: Mohammadi, F., &#38; Moradi, S. (2015). Resolution of unmixed bipartite graphs.
    <i>Bulletin of the Korean Mathematical Society</i>. Korean Mathematical Society.
    <a href="https://doi.org/10.4134/BKMS.2015.52.3.977">https://doi.org/10.4134/BKMS.2015.52.3.977</a>
  chicago: Mohammadi, Fatemeh, and Somayeh Moradi. “Resolution of Unmixed Bipartite
    Graphs.” <i>Bulletin of the Korean Mathematical Society</i>. Korean Mathematical
    Society, 2015. <a href="https://doi.org/10.4134/BKMS.2015.52.3.977">https://doi.org/10.4134/BKMS.2015.52.3.977</a>.
  ieee: F. Mohammadi and S. Moradi, “Resolution of unmixed bipartite graphs,” <i>Bulletin
    of the Korean Mathematical Society</i>, vol. 52, no. 3. Korean Mathematical Society,
    pp. 977–986, 2015.
  ista: Mohammadi F, Moradi S. 2015. Resolution of unmixed bipartite graphs. Bulletin
    of the Korean Mathematical Society. 52(3), 977–986.
  mla: Mohammadi, Fatemeh, and Somayeh Moradi. “Resolution of Unmixed Bipartite Graphs.”
    <i>Bulletin of the Korean Mathematical Society</i>, vol. 52, no. 3, Korean Mathematical
    Society, 2015, pp. 977–86, doi:<a href="https://doi.org/10.4134/BKMS.2015.52.3.977">10.4134/BKMS.2015.52.3.977</a>.
  short: F. Mohammadi, S. Moradi, Bulletin of the Korean Mathematical Society 52 (2015)
    977–986.
date_created: 2018-12-11T11:52:39Z
date_published: 2015-05-31T00:00:00Z
date_updated: 2021-01-12T06:51:31Z
day: '31'
department:
- _id: CaUh
doi: 10.4134/BKMS.2015.52.3.977
intvolume: '        52'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: http://arxiv.org/abs/0901.3015
month: '05'
oa: 1
oa_version: Preprint
page: 977 - 986
publication: Bulletin of the Korean Mathematical Society
publication_identifier:
  eissn:
  - 2234-3016
publication_status: published
publisher: Korean Mathematical Society
publist_id: '5624'
quality_controlled: '1'
scopus_import: 1
status: public
title: Resolution of unmixed bipartite graphs
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 52
year: '2015'
...