A classical theorem in Combinatorial Optimization proves the existence of fully polynomial- time approximation schemes for the knapsack problem. In a recent paper, Van Vyve and Wolsey ask whether for each 0 < epsilon ≤ 1 there exists an extended formulation for the knapsack problem, of size polynomial in the number of variables and/or 1/epsilon ... Read more
In the seminal work [1] L. Lovász introduced the concept of an orthonormal representation of a graph, and also a related value, now popularly known as the Lováasz number of the graph. One of the remarkable properties of the Lovász number is that it lies sandwiched between the stability number and the complementer chromatic number. … Read more
Adverse drug reactions (ADRs) are estimated to be one of the leading causes of death. Many national and international agencies have set up databases of ADR reports for the express purpose of determining the relationship between drugs and adverse reactions that they cause. We formulate the drug-reaction relationship problem as a continuous optimization problem and … Read more
We prove that cutting-plane proofs which use split cuts have exponential length in the worst case. Split cuts, defined by Cook, Kannan, Schrijver (1993), are known to be equivalent to a number of other classes of cuts, namely mixed-integer rounding (MIR) cuts, Gomory mixed-integer cuts, and disjunctive cuts. Our result thus implies the exponential worst-case … Read more
This paper addresses a constrained two-dimensional (2D) non-guillotine cutting problem, where a fixed set of small rectangles has to be cut from a larger stock rectangle so as to maximize the value of the rectangles cut. The algorithm we propose hybridizes a novel placement procedure with a genetic algorithm based on random keys. We propose … Read more
This paper is the first of a series of two devoted to the polyhedral study of a strongly NP-hard resource-constrained scheduling problem, referred to as the process move programming problem. This problem arises in relation to the operability of certain high-availability real time distributed systems. After a brief introduction to the problem as well as … Read more
This paper is the second of a series of two devoted to the polyhedral study of a strongly NP-hard resource-constrained scheduling problem, referred to as the process move programming problem. In the present paper, we put back into the picture the capacity constraints which were ignored in the first paper. In doing so, we introduce … Read more
We consider k-regular graphs with loops, and study the Lovász theta-numbers and Schrijver theta’-numbers of the graphs that result when the loop edges are removed. We show that the theta-number dominates a recent eigenvalue upper bound on the stability number due to Godsil and Newman [C.D. Godsil and M.W. Newman. Eigenvalue bounds for independent sets. … Read more
Continuous GRASP (C-GRASP) is a stochastic local search metaheuristic for finding cost-efficient solutions to continuous global optimization problems subject to box constraints (Hirsch et al., 2006). Like a greedy randomized adaptive search procedure (GRASP), a C-GRASP is a multi-start procedure where a starting solution for local improvement is constructed in a greedy randomized fashion. In … Read more
The aim of this paper is to study how efficiently we evaluate a system of multivariate polynomials and their partial derivatives in homotopy continuation methods. Our major tool is an extension of the Hornor scheme, which is popular in evaluating a univariate polynomial, to a multivariate polynomial. But the extension is not unique, and there … Read more