Shattering Inequalities for Learning Optimal Decision Trees

Recently, mixed-integer programming (MIP) techniques have been applied to learn optimal decision trees. Empirical research has shown that optimal trees typically have better out-of-sample performance than heuristic approaches such as CART. However, the underlying MIP formulations often suffer from slow runtimes, due to weak linear programming (LP) relaxations. In this paper, we first propose a … Read more

Inefficiency of pure Nash equilibria in series-parallel network congestion games

We study the inefficiency of pure Nash equilibria in symmetric unweighted network congestion games defined over series-parallel networks. We introduce a quantity y(D) to upper bound the Price of Anarchy (PoA) for delay functions in class D. When D is the class of polynomial functions with highest degree p, our upper bound is 2^{p+1} − … Read more

The Price of Anarchy in Series-Parallel Network Congestion Games

We study the inefficiency of pure Nash equilibria in symmetric network congestion games defined over series-parallel networks with affine edge delays. For arbitrary networks, Correa (2019) proved a tight upper bound of 5/2 on the PoA. On the other hand, for extension-parallel networks, a subclass of series-parallel networks, Fotakis (2010) proved that the PoA is … Read more

Short simplex paths in lattice polytopes

We consider the problem of optimizing a linear function over a lattice polytope P contained in [0,k]^n and defined via m linear inequalities. We design a simplex algorithm that, given an initial vertex, reaches an optimal vertex by tracing a path along the edges of P of length at most O(n^6 k log k). The … Read more

Tight cycle relaxations for the cut polytope

We study the problem of optimizing an arbitrary weight function w’z over the metric polytope of a graph G=(V,E), a well-known relaxation of the cut polytope. We define the signed graph (G, E^-), where E^- consists of the edges of G having negative weight. We characterize the sign patterns of the weight vector w such … Read more

Totally Unimodular Congestion Games

We investigate a new class of congestion games, called Totally Unimodular Congestion Games, in which the strategies of each player are expressed as binary vectors lying in a polyhedron defined using a totally unimodular constraint matrix and an integer right-hand side. We study both the symmetric and the asymmetric variants of the game. In the … Read more

How tight is the corner relaxation? Insights gained from the stable set problem

The corner relaxation of a mixed-integer linear program is a central concept in cutting plane theory. In a recent paper Fischetti and Monaci provide an empirical assessment of the strength of the corner and other related relaxations on benchmark problems. In this paper we give a precise characterization of the bounds given by these relaxations … Read more