Abstract The Weighted Safe Set Problem requires to partition an undirected graph into two families of connected components, respectively denoted as safe and unsafe, in such a way that each safe component dominates the unsafe adjacent components with respect to a weight function. We introduce a combinatorial branch and bound approach, whose main strength is a refined relaxation that combines graph manipulations and the solution of an auxiliary problem. We also propose fixing procedures to reduce the number of branching nodes. The algorithm solves all weighted instances available in the literature and most unweighted ones, up to 50 vertices, with computational times orders of magnitude smaller than the competing algorithms. In order to investigate the limits of the approach, we introduce a benchmark of graphs with 60 vertices, solving to optimality the denser instances.

Abstract We tackle a stochastic version of the critical node problem (CNP) where the goal is to minimize the pairwise connectivity of a graph by attacking a subset of its nodes. In the stochastic setting considered, the outcome of attacks on nodes is uncertain. In our work, we focus on trees and demonstrate that over trees the stochastic CNP actually generalizes to the stochastic critical element detection problem where the outcome of attacks on edges is also uncertain. We prove the NP‐completeness of the decision version of the problem when connection costs are one, while its deterministic counterpart was proved to be polynomial. We then derive a nonlinear model for the considered CNP version over trees and provide a corresponding linearization based on the concept of probability chains. Moreover, given the features of the derived linear model, we devise an exact Benders decomposition (BD) approach where we solve the slave subproblems analytically. A strength of our approach is that it does not rely on any statistical approximation such as the sample average approximation, which is commonly employed in stochastic optimization. We also introduce an approximation algorithm for the problem variant with unit connection costs and unit attack costs, and a specific integer linear model for the case where all the survival probabilities of the nodes in case of an attack are equal. Our methods are capable of solving relevant instances of the problem with hundreds of nodes within 1 hour of computational time. With this work, we aim to foster research on stochastic versions of the CNP, a problem tackled mainly in deterministic contexts so far. Interestingly, we also show a successful application of the concept of probability chains for problem linearizations significantly improved by decomposition methods such as the BD.

We study a prize-collecting single-machine scheduling problem with hard deadlines, where the objective is to minimize the difference between the total tardiness and the total prize of the selected jobs. This problem is motivated by industrial applications, both as a stand-alone model and as a pricing subproblem in column-generation algorithms for parallel machine scheduling problems. A preprocessing rule is devised to identify jobs that cannot belong to any optimal schedule. The resulting reduced problem is solved to optimality by a branch-and-bound algorithm and two integer linear programming formulations. The algorithm and the formulations are experimentally compared on randomly generated benchmark instances.

We present two metaheuristics for the Critical Node Problem, that is, the maximal fragmentation of a graph through the deletion of 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. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 67(3), 209–221 2016