We study quantitative criteria for evaluating the strength of valid inequalities for Gomory and Johnson’s finite and infinite group models and we describe the valid inequalities that are optimal for these criteria. We justify and focus on the criterion of maximizing the volume of the nonnegative orthant cut off by a valid inequality. For the … Read more
In a recent paper, Basu, Hildebrand, and Molinaro established that the set of continuous minimal functions for the 1-dimensional Gomory-Johnson infinite group relaxation possesses a dense subset of extreme functions. The n-dimensional version of this result was left as an open question. In the present paper, we settle this query in the affirmative: for any … Read more
In this paper, we present lower bounds on the rank of the split closure, the multi-branch closure and the lattice-free closure for packing sets as a function of the integrality gap. We also provide a similar lower bound on the split rank of covering polyhedra. These results indicate that whenever the integrality gap is high, … Read more
Despite recent interest in multiobjective integer programming, few algorithms exist for solving biobjective mixed integer programs. We present such an algorithm: the Boxed Line Method. For one of its variants, we prove that the number of single-objective integer programs solved is bounded by a linear function of the number of nondominated line segments in the … Read more
The minimum k-partition problem is a challenging combinatorial problem with a diverse set of applications ranging from telecommunications to sports scheduling. It generalizes the max-cut problem and has been extensively studied since the late sixties. Strong integer formulations proposed in the literature suffer from a prohibitive number of valid inequalities and integer variables. In this … Read more
We investigate methods to solve the maximum-reliability stochastic network interdiction problem (SNIP). In this problem, a defender interdicts arcs on a directed graph to minimize an attacker’s probability of undetected traversal through the network. The attacker’s origin and destination are unknown to the defender and assumed to be random. SNIP can be formulated as a … Read more
We present novel mixed-integer programming (MIP) formulations for (nonconvex) piecewise linear functions. Leveraging recent advances in the systematic construction of MIP formulations for disjunctive sets, we derive new formulations for univariate functions using a geometric approach, and for bivariate functions using a combinatorial approach. All formulations derived are small (logarithmic in the number of piecewise … Read more
We address the multi container loading problem of a company that has to serve its customers by first putting the products on pallets and then loading the pallets into trucks. We approach the problem by developing and solving integer linear models. To be useful in practice, our models consider three types of constraints: geometric constraints, … Read more
We study disruptions at a major airport. Disruptions could be caused by bad weather, for example. Our study is from the perspective of the airport, the air services provider (such as air traffic control) and the travelling public, rather than from the perspective of a single airline. Disruptions cause flights to be subjected to ground … Read more
Christofides and Hadjiconstantinou introduced a dynamic programming state space relaxation for obtaining upper bounds for the Constrained Two-dimensional Guillotine Cutting Problem. The quality of those bounds depend on the chosen item weights, they are adjusted using a subgradient-like algorithm. This paper proposes Algorithm X, a new weight adjusting algorithm based on integer programming that provably … Read more