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Strong lower bounds for the prize collecting Steiner problem in graphs

In this paper, we present an integer programming formulation of the prize collecting Steiner problem in graphs (PCSPG) and describe an algorithm to obtain lower bounds for the problem. The algorithm is based on polyhedral cutting planes and is initiated with tests that attempt to reduce the size of the input graph. Computational experiments were carried out to evaluate the strength of the formulation through its linear programming relaxation. On 96 out of the 114 instances tested, integer solutions were found (thus generating optimal PCSPG solutions).

A new formulation for the Traveling Deliveryman Problem

The 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.

Using Lagrangian dual information to generate degree constrained spanning trees

In this paper, a Lagrangian-based heuristic is proposed for the degree constrained minimum spanning tree problem. The heuristic uses Lagrangian relaxation information to guide the construction of feasible solutions to the problem. The scheme operates, within a Lagrangian relaxation framework, with calls to a greedy construction heuristic, followed by a heuristic improvement procedure. A look ahead infeasibility prevention mechanism, introduced into the greedy heuristic, allowed us to solve instances of the problem where some of the vertices are restricted to having degrees 1 or 2. Furthermore, in order to cut down on CPU time, a restricted version of the original problem is formulated and used to generate feasible solutions. Extensive computational experiments were conducted and indicate that the proposed heuristic is competitive with the best heuristics and metaheuristics in the literature.

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.

Non Delayed Relax-and-Cut Algorithms

Attempts to allow exponentially many inequalities to be candidates to Lagrangian dualization date from the early 1980s. 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.

Solving Diameter Constrained Minimum Spanning Tree Problems in Dense Graphs

In this study, a lifting procedure is applied to some existing formulations of the Diameter Constrained Minimum Spanning Tree Problem. This problem typically models network design applications where all vertices must communicate with each other at minimum cost, while meeting or surpassing a given quality requirement. An alternative formulation is also proposed for instances of the problem where the diameter of feasible spanning trees can not exceed given odd numbers. This formulation dominated their counterparts in this study, in terms of the computation time required to obtain proven optimal solutions. First ever computational results are presented here for complete graph instances of the problem. Sparse graph instances as large as those found in the literature were solved to proven optimality for the case where diameters can not exceed given odd numbers. For these applications, the corresponding computation times are competitive with those found 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.

Stronger K-tree relaxations for the vehicle routing problem☆

Abstract A Lagrangian based exact solution algorithm for the vehicle routing problem (VRP), defined on an undirected graph, is introduced in this paper. Lower bounds are obtained by allowing exponentially many inequalities as candidates to Lagrangian dualization. Three different families of strong valid inequalities (each one with exponentially many elements) are used within VRP formulations. For each of them, separation procedures are proposed for points that define incidence vectors of K -trees. Violated inequalities identified in this way are then dualized in a relax and cut framework. Upper bounds are generated through a Lagrangian Clarke and Wright heuristic. A variable fixation test based on (approximating) linear programming reduced costs, is also implemented. Computational results are presented for the proposed algorithm.

Optimal rectangular partitions

Assume that a rectangle R is given on the Euclidean plane together with a finite set P of points that are interior to R. A rectangular partition of R is a partition of the surface of R into smaller rectangles. The length of such a partition equals the sum of the lengths for the line segments that define it. The partition is said to be feasible if no point of P is interior to a partition rectangle. The Rectangular Partitioning Problem (RPP) seeks a feasible rectangular partition of R with the least length. Computational evidence from the literature indicates that RPPs with noncorectilinear points in P, denoted NCRPPs, are the hardest to solve to proven optimality. In this paper, some structural properties of optimal feasible NCRPP partitions are presented. These properties allow substantial reductions in problem input size to be carried out. Additionally, a stronger formulation of the problem is also made possible. Based on these ingredients, a hybrid Lagrangian Relaxation—Linear Programming Relaxation exact solution algorithm is proposed. Such an algorithm has proved capable of solving NCRPP instances more than twice as large as those found in 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.