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.

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We study the weighted circuit constraint in the context of constraint programming. It appears as a substructure in many practical applications, particularly routing problems. We propose a domain filtering algorithm for the weighted circuit constraint that is based on the 1-tree relaxation of Held and Karp. In addition, we study domain filtering based on an additive bounding procedure that combines the 1-tree relaxation with the assignment problem relaxation. Experimental results on Traveling Salesman Problem instances demonstrate that our filtering algorithms can dramatically reduce the problem size. In particular, the search tree size and solving time can be reduced by several orders of magnitude, compared to existing constraint programming approaches. Moreover, for medium-size problem instances, our method is competitive with the state-of-the-art special-purpose TSP solver Concorde.