On the Average Behaviour of Primal and Dual Greedy Algorithms for the Knapsack Problem

Abstract

We consider primal (“best-in”) and dual (“worst-out”) greedy algorihms for the knapsack problem with Boolean variables. For the primal greedy algorithms, the worst-case behaviour is well-known (cf. [2]), for the dual greedy algorithm it can be arbitrarily bad. We study the average performance of both algorithms. It is shown (on the basis of the Central Limit Theorem) that in average the objective function value of the dual greedy algorithm differs insignificantly from the objective function value of the linear relaxation (and hence from the primal greedy and the optimal objective function values). This means that both the primal and the dual greedy algorithms are in a certain sense asymptotically optimal. This sharpens the results obtained by the authors in [1].