A partitioning scheme for solving the 0–1 knapsack problem

The application of valid inequalities to provide relaxations which can produce tight bounds, is now common practice in Combinatorial Optimisation. This paper attempts to complement current theoretical investigations in this regard. We experimentally search for “valid” equalities which have the potential of strengthening the problem's formulation. Recently, Martello and Toth included cardinality constraints to derive tight upper bounds for the 0–1 Knapsack Problem. Encouraged by their results, we partition the search space by using equality cardinality constraints. Instead of solving the original problem, an equivalent problem, which consists of one or more 0–1 Knapsack Problem with an exact cardinality bound, is solved. By explicitly including a bound on the cardinality, one is able to reduce the size of each subproblem and compute tight upper bounds. Good feasible solutions found along the way are employed to reduce the computational effort by reducing the number of trees searched and the size of the subsequent search tree. We give a brief description of two Lagrangian-based Branch-and-Bound algorithms for solving the exact cardinality bounded subproblems and report on results of numerical experiments with a sequential implementation. Implications for and strategies towards parallel implementation are also given.