A Polytope for a Product of Real Linear Functions in 0/1 Variables

In the context of integer programming, we develop a polyhedral method for linearizing a product of a pair of real linear functions in 0/1 variables. As an example, by writing a pair of integer variables in binary expansion, we have a technique for linearizing their product. We give a complete linear description for the resulting … Read more

Multiprocessor Scheduling under Precedence Constraints: Polyhedral Results

We consider the problem of scheduling a set of tasks related by precedence constraints to a set of processors, so as to minimize their makespan. Each task has to be assigned to a unique processor and no preemption is allowed. A new integer programming formulation of the problem is given and strong valid inequalities are … Read more

An Exact Algorithm for the Capacitated Vertex p-Center Problem

We develop a simple and practical exact algorithm for the problem of locating $p$ facilities and assigning clients to them within capacity restrictions in order to minimize the maximum distance between a client and the facility to which it is assigned (capacitated $p$-center). The algorithm iteratively sets a maximum distance value within which it tries … Read more

Decomposition and Dynamic Cut Generation in Integer Linear Programming

Decomposition algorithms such as Lagrangian relaxation and Dantzig-Wolfe decomposition are well-known methods that can be used to generate bounds for mixed-integer linear programming problems. Traditionally, these methods have been viewed as distinct from polyhedral methods, in which bounds are obtained by dynamically generating valid inequalities to strengthen the linear programming relaxation. Recently, a number of … Read more

The Quadratic Selective Travelling Saleman Problem

A well-known extension of the Travelling Salesman Problem (TSP) is the Selective TSP (STSP): Each node has an associated profit and instead of visiting all nodes, the most profitable set of nodes, taking into account the tour cost, is visited. The Quadratic STSP (QSTSP) adds the additional complication that each pair of nodes have an … Read more

Cover and pack inequalities for (mixed) integer programming

We review strong inequalities for fundamental knapsack relaxations of (mixed) integer programs. These relaxations are the 0-1 knapsack set, the mixed 0-1 knapsack set, the integer knapsack set, and the mixed integer knapsack set. Our aim is to give a unified presentation of the inequalities based on covers and packs and highlight the connections among … Read more

Lift-and-project ranks and antiblocker duality

Recently, Aguilera et al.\ exposed a beautiful relationship between antiblocker duality and the lift-and-project operator proposed by Balas et al. We present a very short proof of their result that the \BCC-rank of the clique polytope is invariant under complementation. The proof of Aguilera et al. relies on their main technical result, which describes a … Read more

An Efficient Exact Algorithm for the Vertex hBcCenter Problem and Computational Experiments for Different Set Covering Subproblems

We develop a simple and yet very efficient exact algorithm for the problem of locating $p$ facilities and assigning clients to them in order to minimize the maximum distance between a client and the facility to which it is assigned. The algorithm iteratively sets a maximum distance value within which it tries to assign all … Read more

The stable set problem and the lift-and-project ranks of graphs

We study the lift-and-project procedures for solving combinatorial optimization problems, as described by Lov\’asz and Schrijver, in the context of the stable set problem on graphs. We investigate how the procedures’ performances change as we apply fundamental graph operations. We show that the odd subdivision of an edge and the subdivision of a star operations … Read more