No abstract

A polytrope is a tropical polytope which at the same time is convex in the ordinary sense. A d-dimensional polytrope turns out to be a tropical simplex, that is, it is the tropical convex hull of d + 1 points. This statement is equivalent to the known fact that the Segre product of two full polynomial rings (over some field K) has the Gorenstein property if and only if the factors are generated by the same number of indeterminates. The combinatorial types of polytropes up to dimension three are classified.

The tropical Grassmannian parameterizes tropicalizations of ordinary linear spaces, while the Dressian parameterizes all tropical linear spaces in TP n−1 . We study these parameter spaces and we compute them explicitly for n ≤ 7. Planes are identified with matroid subdivisions and with arrangements of trees. These representations are then used to draw pictures.

Abstract. We prove that primal-dual log-barrier interior point methods are not strongly polynomial, by constructing a family of linear programs with 3r + 1 inequalities in dimension 2r for which the number of iterations performed is in \Omega (2 r ). The total curvature of the central path of these linear programs is also exponential in r, disproving a continuous analogue of the Hirsch conjecture proposed by Deza, Terlaky, and Zinchenko. Our method is to tropicalize the central path in linear programming. The tropical central path is the piecewise linear limit of the central paths of parameterized families of classical linear programs viewed through``logarithmic glasses."" This allows us to provide combinatorial lower bounds for the number of iterations and the total curvature, in a general setting.

We develop a tropical analog of the simplex algorithm for linear programming. In particular, we obtain a combinatorial algorithm to perform one tropical pivoting step, including the computation of reduced costs, in O(n(m + n)) time, where m is the number of constraints and n is the dimension.2010 Mathematics Subject Classification. 14T05, 90C05.