Xavier Roy is a theoretical physicist. During his PhD and postdoc, he developed several mathematical models, algorithms and numerical simulations to describe and study perturbations in cosmology and the evolution of the universe.
Project: The knapsack problem
The knapsack problem is a linear problem of combinatory optimization with constraints.
Given a set of n elements i of profits and costs ci, it consists in finding the solutions xi (the number of copies) that maximize the objective function ∑ pi xi while satisfying the constraint ∑ ci xi ⩽ C, with C a given parameter.
The 0-1 knapsack problem restricts the number of copies to 1 (an item i can be either picked (xi = 1) or not (xi = 0)), the bounded knapsack allows for a number of nc copies and the unbounded version for an unlimited number of copies.
Other possible extensions consider multiple objective functions ∑ pi xi → ∑ p ij xi), multiple constraints ( ∑ ci xi ⩽ C → ∑ cij xi ⩽ Cj) or combination of them.
The aim of the project is to explore the knapsack problem, implement in Mathematica an algorithm to solve it efficiently and define a user-friendly function from it.