We show that toric ideals of flow polytopes are generated in degree 3. This was conjectured by Diaconis and Eriksson for the special case of the Birkhoff polytope. Our proof uses a hyperplane subdivision method developed by Haase and Paffenholz. It is known that reduced revlex Gröbner bases of the toric ideal of the Birkhoff polytope Bn have at most degree n. We show that this bound is sharp fo...

We define a centrally symmetric analogue of the cyclic polytope and study its facial structure. We conjecture that our polytopes provide asymptotically the largest number of faces in all dimensions among all centrally symmetric polytopes with n vertices of a given even dimension d = 2k when d is fixed and n grows. For a fixed even dimension d = 2k and an integer 1 ≤ j < k we prove that the maxi...

The Birkhoff polytope (the convex hull of the set of permutation matrices), which is represented using Θ(n) variables and constraints, is frequently invoked in formulating relaxations of optimization problems over permutations. Using a recent construction of Goemans [1], we show that when optimizing over the convex hull of the permutation vectors (the permutahedron), we can reduce the number of...

We prove a bijection between the triangulations of 3-dimensional cyclic polytope C(n+2, 3) and persistent graphs with n vertices. show that under this Stasheff-Tamari orders on naturally translate to subgraph inclusion graphs. Moreover, we describe connection second higher Bruhat order B(n, 2). additionally give an algorithm efficiently enumerate all vertices thus 3).

In 1921 Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that #(K∩Z) ≤ n! vol(K)+n, whenever K ⊂ R is a convex body containing n+1 affinely independent integral points. Here we prove an analogous inequality with respect to the surface area F(K), namely #(K ∩ Z) < vol(K) + ((√n + 1)/2) (n − 1)...

The Birkhoff polytope (the convex hull of the set of permutation matrices) is frequently invoked informulating relaxations of optimization problems over permutations. The Birkhoff polytope is representedusing Θ(n) variables and constraints, significantly more than the n variables one could use to representa permutation as a vector. Using a recent construction of Goemans [1], we ...

The Birkhoff (permutation) polytope, Bn, consists of the n × n nonnegative doubly stochastic matrices, has dimension (n− 1)2, and has n2 facets. A new analogue, the alternating sign matrix polytope, ASMn, is introduced and characterized. Its vertices are the Qn−1 j=0 (3j+1)! (n+j)! n × n alternating sign matrices. It has dimension (n− 1)2, has 4[(n− 2)2 +1] facets, and has a simple inequality d...

We consider symmetric multi-marginal Kantorovich optimal transport problems on finite state spaces with uniform-marginal constraint. These consist of minimizing a linear objective function over high-dimensional polytope, here referred to as polytope. The presented results are split nature, computational and theoretical. Within the part only small numbers marginals $N$ marginal sites $\ell$ cons...

We study the calculation of the volume of the polytope Bn of n × n doubly stochastic matrices; that is, the set of real non-negative matrices with all row and column sums equal to one. We describe two methods. The first involves a decomposition of the polytope into simplices. The second involves the enumeration of “magic squares”, i.e., n×n non-negative integer matrices whose rows and columns a...