The knapsack problem: A survey

A unifying survey of the literature related to the knapsack problem; that is, maximize \documentclass{article}\pagestyle{empty}$ \sum\limits_i {v_i x_{i,} } $ , subject to \documentclass{article}\pagestyle{empty}$ \sum\limits_j {w_i x_i W} $ and x_i ⩾ 0, integer; where v_i, w_i and W are known integers, and w_i (i = 1, 2, …, N) and W are positive. Various uses, including those in group theory and in other integer programming algorithms, as well as applications from the literature, are discussed. Dynamic programming, branch and bound, search enumeration, heuristic methods, and other solution techniques are presented. Computational experience, and extensions of the knapsack problem, such as to the multi‐dimensional case, are also considered.