Alexandre Salles da Cunha

A relax-and-cut algorithm for the prize-collecting Steiner problem in graphs

Given an undirected graph G with penalties associated with its vertices and costs associated with its edges, a Prize Collecting Steiner (PCS) tree is either an isolated vertex of G or else any tree of G, be it spanning or not. The weight of a PCS tree equals the sum of the costs for its edges plus the sum of the penalties for the vertices of G not spanned by the PCS tree. Accordingly, the Prize Collecting Steiner Problem in Graphs (PCSPG) is to find a PCS tree with the lowest weight. In this paper, after reformulating and re-interpreting a given PCSPG formulation, we use a Lagrangian Non Delayed Relax and Cut (NDRC) algorithm to generate primal and dual bounds to the problem. The algorithm is capable of adequately dealing with the exponentially many candidate inequalities to dualize. It incorporates ingredients such as a new PCSPG reduction test, an effective Lagrangian heuristic and a modification in the NDRC framework that allows duality gaps to be further reduced. The Lagrangian heuristic suggested here dominates their PCSPG counterparts in the literature. The NDRC PCSPG lower bounds, most of the time, nearly matched the corresponding Linear Programming relaxation bounds.

The Pickup and Delivery Problem with Cross-Docking

Usual models that deal with the integration of vehicle routing and cross-docking operations impose that every vehicle must stop at the dock even if the vehicle collects and delivers the same set of goods. In order to allow vehicles to avoid the stop at the dock and thus, reduce transportation costs, we introduce the Pickup and Delivery Problem with Cross-Docking (PDPCD). An Integer Programming formulation and a Branch-and-price algorithm for the problem are discussed. Our computational results indicate that optimal or near optimal solutions for PDPCD indeed allow total costs to be significantly reduced. Due to improvements in the resolution of the pricing problems, the Branch-and-price algorithm for PDPCD works better than similar algorithms for other models in the literature.

Benders Decomposition, Branch-and-Cut, and Hybrid Algorithms for the Minimum Connected Dominating Set Problem

We present exact algorithms for solving the minimum connected dominating set problem in an undirected graph. The algorithms are based on two approaches: a Benders decomposition algorithm and a branch-and-cut method. We also develop a hybrid algorithm that combines these two approaches. Two variants of each of the three resulting algorithms are considered: a stand-alone version and an iterative probing variant. The latter variant is based on a simple property of the problem, which states that if no connected dominating set of a given cardinality exists, then there are no connected dominating sets of lower cardinality. We present computational results on a large set of instances from the literature.

Branch-and-price algorithms for the Two-Echelon Capacitated Vehicle Routing Problem

In this paper, we propose an Integer Programming formulation and two branch-and-price implementations for the Two-Echelon Capacitated Vehicle Routing Problem. One algorithm considers routes that satisfy the elementarity condition, while the other relaxes such constraint when pricing routes. For instances that could not be solved to proven optimality within a given time limit, our computational experience suggests that the former provides sharper upper bounds while the latter offers tighter lower bounds. As a by-product, ten new best upper bounds and two new optimality certificates are provided here.

A Branch-and-price algorithm for a Vehicle Routing Problem with Cross-Docking

Abstract n this paper, we propose a reformulation and a Branch-and-price (BP) algorithm for the Vehicle Routing Problem with Cross-Docking (VRPCD). Our computational results indicate that the reformulation provides bounds much stronger than network flow bounds from previous studies. As a consequence, when BP and a Linear Programming based Branch-and-bound (LPBB) method (that relies on the network flow formulation) are run for the same restricted time limit, BP clearly dominates LPBB in terms of the quality of lower and upper bounds found during the search.

Heuristic and exact algorithms for a min-max selective vehicle routing problem

In this work, we investigate a vehicle routing problem where not all clients need to be visited and the goal is to minimize the longest vehicle route. We propose two exact solution approaches for solving the problem: a Branch-and-cut (BC) algorithm and a Local Branching (LB) method that uses BC as its inner solver. Our computational experience indicates that, in practice, the problem is difficult to solve, mainly when the number of vehicles grows. In addition to the exact methods, we present a heuristic that relies on GRASP and on the resolution of a restricted integer program based on a set covering reformulation for the problem. The heuristic was capable of significantly improving the best solutions provided by BC and LB, in one tenth of the times taken by them to achieve their best upper bounds.

A Branch-and-Cut-and-Price Algorithm for the Two-Echelon Capacitated Vehicle Routing Problem

In this paper, we introduce a branch-and-cut-and-price algorithm for the two-echelon capacitated vehicle routing problem. The algorithm relies on a reformulation based on q-routes that combines two important features. First, it overcomes symmetry issues observed in a formulation coming from a previous study of the problem. Second, it is strengthened with several classes of valid inequalities. As a result, the branch-and-cut-and-price implementation compares favorably with previous exact solution approaches for the problem-namely, two branch-and-price algorithms and a branch-and-cut method. Overall, 10 new optimality certificates and 8 new best upper bounds are provided in this study. New best lower bounds are also presented for all instances in the hardest test set from the literature.

The minimum connected dominating set problem: formulation, valid inequalities and a branch-and-cut algorithm

We consider the minimum connected dominating set problem. We present an integer programming formulation and new valid inequalities. A branchand-cut algorithm based on the reinforced formulation is also provided. Computational results indicate that the reinforced lower bounds are always stronger than the bounds implied by the formulation from which resulted one of the best known exact algorithms for the problem. In some cases, the reinforced lower bounds are stronger than those implied by the strongest known formulation to date. For dense graphs, our algorithm provides the best results in the literature. For sparse instances, known to be harder, our method is outperformed by another one. We discuss reasons for that and how to improve our current computational results. One possible way to achieve such goals is to devise specific separation algorithms for some classes of valid inequalities introduced here.

A novel column generation algorithm for the vehicle routing problem with cross-docking

In this paper we present a novel column generation (CG) formulation and a branch-and-price (BP) algorithm for the Vehicle Routing Problem with Cross-Docking (VRPCD). Our BP algorithm is compared with a previous algorithm to solve the VRPCD and the computational results show that our approach dominates the other in terms of the quality of lower and upper bounds and also can evaluate optimal solutions faster.

The k-Cardinality Tree Problem: Reformulations and Lagrangian Relaxation

Given an undirected defining graph for the k-Cardinality Tree Problem (KCTP), an associated directed graph involving two additional vertices is introduced in this paper and gives rise to two compact reformulations of the problem. For the first one, connectivity of feasible solutions is enforced through multicommodity flows while, for the other, lifted Miller-Tucker-Zemlin constraints are used. Comparing the two reformulations, much stronger Linear Programming relaxation bounds are obtained from the first one, albeit at much higher CPU times. However, a Branch-and-Bound algorithm based on the second reformulation proved much more effective and managed to obtain, for the first time, optimality certificates for a large number of KCTP instances from the literature. Additionally, for some instances where optimality could not be proven within the given pre-specified CPU time limit, new best upper bounds were generated. Finally, a Lagrangian heuristic based on the first reformulation was also implemented and proved capable of generating feasible KCTP solutions comparable in quality with the best overall results obtained by metaheuristic based heuristics found in the literature. For our test cases, Lagrangian upper bounds are no more than 3.8% away from the best upper bounds known. Additionally, several new best upper bounds and optimality certificates were obtained by the heuristic. Corresponding Lagrangian heuristic CPU times, however, are typically higher than those associated with their competitors.