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We consider the 0-1 Knapsack Problem with Setups. We propose an exact approach which handles the structure of the ILP formulation of the problem. It relies on partitioning the variables set into two levels and exploiting this partitioning. The proposed approach favorably compares to the algorithms in literature and to solver CPLEX 12.5 applied to the ILP formulation. It turns out to be very effective and capable of solving to optimality, within limited CPU time, all instances with up to 100000 variables.

We present two metaheuristics for the Critical Node Problem, that is, the maximal fragmentation of a graph through the deletion of k nodes. The two metaheuristics are based on the Iterated Local Search and Variable Neighborhood Search frameworks. Their main characteristic is to exploit two smart and computationally efficient neighborhoods which we show can be implemented far more efficiently than the classical neighborhood based on the exchange of any two nodes in the graph, and which we prove is equivalent to the classical neighborhood in the sense that it yields the same set of neighbors. Solutions to improve the overall running time without deteriorating the quality of the solution computed are also illustrated. The results of the proposed metaheuristics outperform those currently available in literature.

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In this paper, we consider the 0–1 knapsack problem with setups. Items are grouped into families and if any items of a family are packed, this induces a setup cost as well as a setup resource consumption. We introduce a new dynamic programming algorithm that performs much better than a previous dynamic program and turns out to be also a valid alternative to an exact approach based on the use of an Integer Linear Programming (ILP) solver. Then we present a general inapproximability result. Furthermore, we investigate several relevant special cases that still permit fully polynomial‐time approximation schemes and others where the problem remains hard to approximate.