Recently a generalization of simple convex polytopes to combinatorial entit ies known as abstract polytopes has been proposed. The graph of an abstract polytope o f d imension d is a regular connected graph of degree d. Given a connected regular graph F o f degree d, it is interesting to find out whether it is the graph of some abstract polytope P. We obtain necessary and sufficient condit ions...

We introduce a globally-convergent algorithm for optimizing the tree-reweighted (TRW) variational objective over the marginal polytope. The algorithm is based on the conditional gradient method (Frank-Wolfe) and moves pseudomarginals within the marginal polytope through repeated maximum a posteriori (MAP) calls. This modular structure enables us to leverage black-box MAP solvers (both exact and...

In this work we present a procedure for automatic parallel code generation in the case of algorithms described through Set of Affine Recurrence Equations (SARE); starting from the original SARE description in an N-dimensional iteration space, the algorithm is converted into a parallel code for an m-dimensional distributed memory parallel machine (m < N). The used projection technique is based o...

We study the facial structure of two important permutation polytopes in R n 2 , the Birkho or assignment polytope B n , de ned as the convex hull of all n n permutation matrices, and the asymmetric traveling salesman polytope T n , de ned as the convex hull of those n n permutation matrices corresponding to n-cycles. Using an isomorphism between the face lattice of B n and the lattice of elemen...

We propose a cutting-plane style algorithm for finding the maximum a posteriori (MAP) state and approximately inferring marginal probabilities in discrete Markov Random Fields (MRFs). The variational formulation of both problems consists of an optimization over the marginal polytope, with the latter having an additional non-linear entropy term in the objective. While there has been significant ...

If ∆ is a polytope in real affine space, each edge of ∆ determines a reflection in the perpendicular bisector of the edge. The exchange group W (∆) is the group generated by these reflections, and ∆ is a (Coxeter) matroid polytope if this group is finite. This simple concept of matroid polytope turns out to be an equivalent way to define Coxeter matroids. The GelfandSerganova Theorem and the st...