A statistical analysis of the knapsack problem

Abstract

We investigate the dependence of the multi-knapsack objective function on the knapsack capacities and on the number of capacity constraints P, in the case when all N objects are assigned the same profit value and the weights are uniformly distributed over the unit interval. A rigorous upper bound to the optimal profit is obtained employing the annealed approximation and then compared with the exact value obtained through the Lagrangian relaxation method. The analysis is restricted to the regime where N goes to infinity and P remains finite.