We investigate a new class of congestion games, called Totally Unimodular (TU) Congestion Games, where the players’ strategies are binary vectors inside polyhedra defined by totally unimodular constraint matrices. Network congestion games belong to this class. In the symmetric case, when all players have the same strategy set, we design an algorithm that finds an optimal aggregated strategy and...

We consider totally unimodular (TU) stochastic programs, that is, two-stage stochastic programs whose extensive-form constraint matrix is TU. We generalize the notion of total unimodularity to apply to sets of matrices and provide properties of such sets. We provide several sufficient conditions on stochastic programs to be TU. When solving TU stochastic problems using the L-shaped method, it i...

We show that finding minimally intersecting n paths from s to t in a directed graph or n perfect matchings in a bipartite graph can be done in polynomial time. This holds more generally for unimodular set systems.

This paper describes methods for counting the number of non-negative integer solutions of the system Ax = b when A is a non-negative totally unimodular matrix and b an integral vector of fixed dimension. The complexity (under a unit cost arithmetic model) is strong in the sense that it depends only on the dimensions of A and not on the size of the entries of b. For the special case of “continge...

We study families of integer circulant matrices and methods for determining which are unimodular. This problem arises in the study of cyclically presented groups, and leads to the following problem concerning polynomials with integer coefficients: given a polynomial f(x) ∈ Z[x], determine all those n ∈ N such that Res(f(x), xn − 1) = ±1. In this paper we describe methods for resolving this prob...

We study algebraic algorithms for expressing the number of non-negative integer solutions to a unimodular system of linear equations as a function of the right hand side. Our methods include Todd classes of toric varieties via Gröbner bases, and rational generating functions as in Barvinok’s algorithm. We report polyhedral and computational results for two special cases: counting contingency ta...