Exact algorithms for the 0-1 Time-bomb Knapsack Problem

We consider a stochastic version of the 0–1 Knapsack Problem in which, in addition to profit and weight, each item is associated with a probability of exploding and destroying all the contents of the knapsack. The objective is to maximize the expected profit of the selected items. The resulting problem, denoted as 0–1 Time-Bomb Knapsack … Read more

Complexity, Exactness, and Rationality in Polynomial Optimization

We study representation of solutions and certificates to quadratic and cubic optimization problems. Specifically, we focus on minimizing a cubic function over a polyhedron and also minimizing a linear function over quadratic constraints. We show when there exist rational feasible solutions (and if we can detect them) and when we can prove feasibility of sublevel … Read more

New algorithms for hierarchical optimisation in kidney exchange programmes

Kidney exchange programmes (KEPs) across the world help match donors and recipients to identify kidney transplantations. Almost all KEPs use a hierarchical set of objectives to determine an optimal set of transplants to perform, and integer linear programming is often used to find such optimal matchings. In this work, we identify the barriers in existing … Read more

Stability in the the Hospitals / Residents problem with Couples and Ties: Mathematical models and computational studies

In the well-known Hospitals/Residents problem (HR), the objective is to find a stable matching of doctors (or residents) to hospitals based on their preference lists. In this paper, we study HRCT, the extension of HR in which doctors are allowed to apply in couples, and in which doctors and hospitals can include ties in their … Read more

Mathematical models for stable matching problems with ties and incomplete lists

We present new integer linear programming (ILP) models for NP-hard optimisation problems in instances of the Stable Marriage problem with Ties and Incomplete lists (SMTI) and its many-to-one generalisation, the Hospitals / Residents problem with Ties (HRT). These models can be used to efficiently solve these optimisation problems when applied to (i) instances derived from … Read more

MSEA.jl: A Multi-Stage Exact Algorithm for Bi-objective Pure Integer Linear Programming in Julia

We present a new exact method for bi-objective pure integer linear programming, the so-called Multi-Stage Exact Algorithm (MSEA). The method combines several existing exact and approximate algorithms in the literature to compute the entire nondominated frontier of any bi-objective pure integer linear program. Each algorithm available in MSEA has multiple versions in the literature. Hence, … Read more

The Vertex k-cut Problem

Given an undirected graph G = (V, E), a vertex k-cut of G is a vertex subset of V the removing of which disconnects the graph in at least k connected components. Given a graph G and an integer k greater than or equal to two, the vertex k-cut problem consists in finding a vertex … Read more

Exact Approaches for the Knapsack Problem with Setups

We consider a generalization of the knapsack problem in which items are partitioned into classes, each characterized by a fixed cost and capacity. We study three alternative Integer Linear Programming formulations. For each formulation, we design an efficient algorithm to compute the linear programming relaxation (one of which is based on Column Generation techniques). We … Read more

Solving rank-constrained semidefinite programs in exact arithmetic

We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active, this is a non-convex optimization problem, otherwise it is a semidefinite program. Both find numerous applications especially in … Read more

The Quest for Optimal Solutions for the Art Gallery Problem: a Practical Iterative Algorithm

The general Art Gallery Problem (AGP) consists in finding the minimum number of guards sufficient to ensure the visibility coverage of an art gallery represented by a polygon. The AGP is a well known NP-hard problem and, for this reason, all algorithms proposed so far to solve it are unable to guarantee optimality except in … Read more