We propose a numerically exact algorithm for solving the Bin-Packing Problem (BPP) based on a branch-price-and-cut framework combined with a pattern-enumeration method. Key to the algorithm is a novel technique for the computation of numerically safe dual bounds for the widely adopted set covering reformulation of the BPP (tightened with additional valid inequalities) with a precision that is higher than the one of general-purpose floating-point solvers. Our branch-price-and-cut algorithm also relies on an exact integer (fixed-point) label setting algorithm for solving the pricing problem associated with the tightened set-covering formulation. To the best of our knowledge, ours is the first algorithm for the BPP that is numerically exact and practical for solving large-scale instances. Extensive computational results on instances affected by notorious numerical difficulties (those of the Augmented Non-IRUP class) show that our exact algorithm outperforms all of the not numerically exact state-of-the-art algorithms based on branch-and-cut-and-price techniques that rely on a set-covering formulation of the BPP. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms − Discrete.

Given a directed graph with capacities and interdiction costs associated with its arcs, the maximum flow blocker problem (MFBP) asks to find a minimum-cost subset of arcs to be removed from the graph in such a way that the remaining maximum-flow value does not exceed a given threshold. The MFBP has applications in telecommunication networks and in the monitoring of civil infrastructures, among others. We propose an integer linear programming formulation (ILP) with an exponential number of constraints, called Benders cut, for the MFBP. Accordingly, we derive a branch-and-cut algorithm to optimally solve the problem. Preliminary experimental results are reported to assess performance of the formulation and more precisely to determine the dimension of the problem that could be solved to proven optimality.

We study the family of problems of partitioning and covering a graph into/with a minimum number of relaxed cliques. Relaxed cliques are subsets of vertices of a graph for which a clique-defining property—for example, the degree of the vertices, the distance between the vertices, the density of the edges, or the connectivity between the vertices—is relaxed. These graph partitioning and covering problems have important applications in many areas such as social network analysis, biology, and disease-spread prevention. We propose a unified framework based on branch-and-price techniques to compute optimal decompositions. For this purpose, new, effective pricing algorithms are developed, and new branching schemes are invented. In extensive computational studies, we compare several algorithmic designs, such as structure-preserving versus dichotomous branching, and their interplay with different pricing algorithms. The final chosen branch-and-price setup produces results that demonstrate the effectiveness of all components of the newly developed framework and the validity of our approach when applied to social network instances.

Given an undirected graph, the Vertex Coloring Problem (VCP) consists of assigning a color to each vertex of the graph in such a way that two adjacent vertices do not share the same color and the total number of colors is minimized. DSATUR‐based Branch‐and‐Bound algorithm (DSATUR) is an effective exact algorithm for the VCP. One of its main drawback is that a lower bound is computed only once and it is never updated. We introduce a reduced graph which allows the computation of lower bounds at nodes of the branching tree. We compare the effectiveness of different classical VCP bounds, plus a new lower bound based on the ‐to‐ mapping between VCPs and Stable Set Problems. Our new DSATUR outperforms the state of the art for random VCP instances with high density, significantly increasing the size of instances solved to proven optimality. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 69(1), 124–141 2017

We propose a framework to model general guillotine restrictions in two-dimensional cutting problems formulated as mixed-integer linear programs (MIPs). The modeling framework requires a pseudopolynomial number of variables and constraints, which can be effectively enumerated for medium-size instances. Our modeling of general guillotine cuts is the first one that, once it is implemented within a state-of-the-art MIP solver, can tackle instances of challenging size. We mainly concentrate our analysis on the guillotine two-dimensional knapsack problem (G2KP), for which a model, and an exact procedure able to significantly improve the computational performance, are given. We also show how the modeling of general guillotine cuts can be extended to other relevant problems such as the guillotine two-dimensional cutting stock problem and the guillotine strip packing problem (GSPP). Finally, we conclude the paper discussing an extensive set of computational experiments on G2KP and GSPP benchmark instances from the literature.

We consider a generalization of the 0–1 knapsack problem in which the profit of each item can take any value in a range characterized by a minimum and a maximum possible profit. A set of specific profits is called a scenario. Each feasible solution associated with a scenario has a regret, given by the difference between the optimal solution value for such scenario and the value of the considered solution. The interval min–max regret knapsack problem (MRKP) is then to find a feasible solution such that the maximum regret over all scenarios is minimized. The problem is extremely challenging both from a theoretical and a practical point of view. Its decision version is complete for the second level of the polynomial hierarchy hence it is most probably not in 𝒩𝒫. In addition, even computing the regret of a solution with respect to a scenario requires the solution of an 𝒩𝒫-hard problem. We examine the behavior of classical combinatorial optimization approaches when adapted to the solution of the MRKP. We introduce an iterated local search approach and a Lagrangian-based branch-and-cut algorithm and evaluate their performance through extensive computational experiments.

Given an indirected graph G = (V;E), a Vertex k-Separator is a subset of the vertex set V such that, when the separator is removed from the graph, the remaining vertices can be partitioned into k subsets that are pairwise edge-disconnected. In this paper we focus on the Balanced Vertex k-Separator Problem, i.e., the problem of finding a minimum cardinality separator such that the sizes of the resulting disconnected subsets are balanced. We present a compact Integer Linear Programming formulation for the problem, and present a polyhedral study of the associated polytope. We also present an Exponential-Size formulation, for which we derive a column generation and a branching scheme. Preliminary computational results are reported comparing the performance of the two formulations on a set of benchmark instances.

We study a natural generalization of the knapsack problem, in which each item exists only for a given time interval. One has to select a subset of the items (as in the classical case), guaranteeing that for each time instant, the set of existing selected items has total weight no larger than the knapsack capacity. We focus on the exact solution of the problem, noting that prior to our work, the best method was the straightforward application of a general-purpose solver to the natural integer linear programming formulation. Our results indicate that much better results can be obtained by using the same general-purpose solver to tackle a nonstandard Dantzig-Wolfe reformulation in which subproblems are associated with groups of constraints. This is also interesting because the more natural Dantzig-Wolfe reformulation of single constraints performs extremely poorly in practice.