@article{1092,
abstract = {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 = {Mohammadi, Fatemeh and Uhler, Caroline and Wang, Charles and Yu, Josephine},
journal = {SIAM Journal on Discrete Mathematics},
number = {1},
pages = {64--93},
publisher = {SIAM},
title = {{Generalized permutohedra from probabilistic graphical models}},
doi = {10.1137/16M107894X},
volume = {32},
year = {2018},
}
@article{1547,
abstract = {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 = {Mohammadi, Fatemeh and Moradi, Somayeh},
issn = {2234-3016},
journal = {Bulletin of the Korean Mathematical Society},
number = {3},
pages = {977 -- 986},
publisher = {Korean Mathematical Society},
title = {{Resolution of unmixed bipartite graphs}},
doi = {10.4134/BKMS.2015.52.3.977},
volume = {52},
year = {2015},
}