Solving the 0–1 proportional knapsack problem by sampling

In this paper, we deal with the proportional knapsack problem that is a variation on the ordinary knapsack problem. In the proportional knapsack problem, we look at filling an urn with objects having two characteristics: color and weight. The colors of the objects in the urn should be proportional to the distribution of the colors in the object universe, and the total weight of the objects in the urn should be as close as possible to the capacity of the urn. The formulation of the problem was motivated by a real-life application from the area of finance, called a dollar roll. We show that the proportional knapsack problem is NP-hard, and then, using sampling, develop a heuristic procedure for solving the problem.Partial support from the Fund for the Promotion of Research and from the Alexander Goldberg Memorial Fund at Technion is gratefully acknowledged.