We consider a scheme to derive lower bounds for the time-dependent traveling salesman problem. It involves splitting lower bounds into a number of components and optimizing each of these components. The lower bounds thus derived are shown to be at least as sharp as the ones previously suggested for the problem. We describe a branch-andbound algorithm based on our lower bounding scheme and computationally test it for an instance of the problem known as the traveling deliveryman 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.

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Given a graph G = (V, E), the maximum leaf spanning tree problem (MLSTP) is to find a spanning tree of G with as many leaves as possible. The problem is easy to solve when G is complete. However, for the general case, when the graph is sparse, it is proven to be NP-hard. In this paper, two reformulations are proposed for the problem. The first one is a reinforced directed graph version of a formulation found in the literature. The second recasts the problem as a Steiner arborescence problem over an associated directed graph. Branch-and-Cut algorithms are implemented for these two reformulations. Additionally, we also implemented an improved version of a MLSTP Branch-and-Bound algorithm, suggested in the literature. All of these algorithms benefit from pre-processing tests and a heuristic suggested in this paper. Computational comparisons between the three algorithms indicate that the one associated with the first reformulation is the overall best. It was shown to be faster than 123 290 A. Lucena et al. the other two algorithms and is capable of solving much larger MLSTP instances than previously attempted in the literature.

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a b s t r a c tThe Traveling Deliveryman Problem is a generalization of the Minimum Cost Hamiltonian Path Problem where the starting vertex of the path, i.e. a depot vertex, is fixed in advance and the cost associated with a Hamiltonian path equals the sum of the costs for the layers of paths (along the Hamiltonian path) going from the depot vertex to each of the remaining vertices. In this paper, we propose a new Integer Programming formulation for the problem and computationally evaluate the strength of its Linear Programming relaxation. Computational results are also presented for a cutting plane algorithm that uses a number of valid inequalities associated with the proposed formulation. Some of these inequalities are shown to be facet defining for the convex hull of feasible solutions to that formulation. These inequalities proved very effective when used to reinforce Linear Programming relaxation bounds, at the nodes of a Branch and Bound enumeration tree.

Attempts to allow exponentially many inequalities to be candidates to Lagrangian dualization date from the early 1980's. In this paper, the term Relax-and-Cut, introduced elsewhere, is used to denote the whole class of Lagrangian Relaxation algorithms where Lagrangian bounds are attempted to be improved by dynamically strengthening relaxations with the introduction of valid constraints. An algorithm in that class, denoted here Non Delayed Relax-and-Cut, is described in detail, together with a general framework to obtain feasible integral solutions. Specific implementations of NDRC are presented for the Steiner Tree Problem and for a Cardinality Constrained Set Partitioning Problem.