Luis Gouveia

Multicommodity flow models for spanning trees with hop constraints

Abstract In this paper we compare the linear programming relaxations of undirected and directed multicommodity flow formulations for the terminal layout problem with hop constraints. Hop constraints limit the number of hops (links) between the computer center and any terminal in the network. These constraints model delay constraints since a smaller number of hops decreases the maximum delay transmission time in the network. They also model reliability constraints because with a smaller number of hops there is a lower route loss probability. Hop constraints are easily modelled with the variables involved in multicommodity flow formulations. We give some empirical evidence showing that the linear programming relaxation of such formulations give sharp lower bounds for this hop constrained network design problem. On the other hand, these formulations lead to very large linear programming models. Therefore, for bounding purposes we also derive several lagrangean based procedures from a directed multicommodity flow formulation and present some computational results taken from a set of instances with up to 40 nodes.

A classification of formulations for the (time-dependent) traveling salesman problem

Abstract The time-dependent traveling salesman problem (TDTSP) is a generalization of the classical traveling salesman problem where the cost of any given arc is dependent of its position in the tour. The TDTSP can model several real world applications (e.g., one-machine sequencing). In this paper we present a classification of formulations for the TDTSP. This framework includes both new and old formulations. The new formulation presented in this paper is derived from a quadratic assignment model for the TDTSP. In a first step, Lawlers transformation procedure is used to derive an equivalent linearized version of the quadratic model. In a second step, a stronger formulation is obtained by tightening some constraints of the previous formulation. It is shown that, in terms of linear relaxations, the latter formulation is either equivalent or better than other formulations already known from the literature. Finally, we compare these formulations with other well known formulations for the classical traveling salesman problem.

Modeling hop-constrained and diameter-constrained minimum spanning tree problems as Steiner tree problems over layered graphs

The hop-constrained minimum spanning tree problem (HMSTP) is an NP-hard problem arising in the design of centralized telecommunication networks with quality of service constraints. We show that the HMSTP is equivalent to a Steiner tree problem (STP) in an appropriate layered graph. We prove that the directed cut model for the STP defined in the layered graph, dominates the best previously known models for the HMSTP. We also show that the Steiner directed cuts in the extended layered graph space can be viewed as being a stronger version of some previously known HMSTP cuts in the original design space. Moreover, we show that these strengthened cuts can be combined and projected into new families of cuts, including facet defining ones, in the original design space. We also adapt the proposed approach to the diameter-constrained minimum spanning tree problem (DMSTP). Computational results with a branch-and-cut algorithm show that the proposed method is significantly better than previously known methods on both problems.

A 2n Constraint Formulation for the Capacitated Minimal Spanning Tree Problem

In this paper we present a new formulation for the Capacitated Minimal Spanning Tree CMST problem. One advantage of the new formulation is that it is more compact in the number of constraints than a well-known formulation. Additionally, we show that the linear programming relaxation of both formulations produces optimal solutions with the same cost. We present a brief discussion concerning valid inequalities for the CMST which are directly derived from the new formulation. We show that some of the new inequalities are not dominated by some sets of facet-inducing inequalities for the CMST. We derive some Lagrangian relaxation-based methods from the new formulation and present computational evidence showing that reasonable improvements on the original linear programming bounds can be obtained if these methods are strengthened by the use of cutting planes. The reported computational results indicate that one of the methods presented in this paper dominates, in most of the cases, the previous best methods reported in the literature. The most significant improvements are obtained in the instances with the root in the corner.

A result on projection for the vehicle routing ptoblem

Abstract In this paper we present a result on projection for the Vehicle Routing Problem (VRP). The VRP is closely related to delivery-type problems and appears in a large number of practical situations concerning the distribution of commodities. The present work focuses on a commodity flow formulation presented by Gavish and Graves. This formulation includes two sets of variables and, hence, it also must include coupling constraints between the two sets of variables. These coupling constraints can be defined in several ways. The main result of this work establishes that when the strongest form of the coupling constraints is used in the flow formulation, the equivalent formulation using only the X ij variables satisfies the so called multistar constraints which, for certain parameters, induce facets of the non-directed VRP polytope. Using an idea taken from Gouveia, we show how to derive a more compact representation, in the number of constraints, of the multistar constraints. Some consequences of our projection result are also discussed.

Using the Miller-Tucker-Zemlin constraints to formulate a minimal spanning tree problem with Hop constraints

Abstract We present several node-oriented formulations for a Hop Constrained Minimal Spanning Tree (HMST) problem. These formulations are based on the well known Miller-Tucker-Zemlin [1] subtour elimination constraints. Liftings to these inequalities are ‘borrowed’ from the literature and a new class of liftings for the same constraints is also presented. We show that the proposed liftings are not dominated by the previously known liftings. We present some lower bounding schemes for the HMST which are based on lagrangean relaxation combined with subgradient optimization. We present some computational results taken from a set of complete graphs with up to 40 nodes.

Using Variable Redefinition for Computing Lower Bounds for Minimum Spanning and Steiner Trees with Hop Constraints

We use variable redefinition (see R. MARTIN, 1987. Generating Alternative Mixed-Integer Programming Models Using Variable Redefinition, Operations Research 35, 820Â 831) to strengthen a multicommodity flow (MCF) model for minimum spanning and Steiner trees with hop constraints between a root node and any other node. Hop constraints model quality of service constraints. The Lagrangean dual value associated with one Lagrangean relaxation derived from the MCF formulation dominates the corresponding LP value. However, the lower bounds given after a reasonable number of iterations of the associated subgradient optimization procedure are, for several cases, still far from the theoretical best limit. Martins variable redefinition technique is used to obtain a generalization of the MCF formulation whose LP bound is equal to the previously mentioned Lagrangean dual bound. We use a set of instances with up to 100 nodes, 50 basic nodes, and 350 edges for comparing an LP approach based on solving the LP relaxation of the new model with the equivalent Lagrangean scheme derived from MCF.

MPLS over WDM network design with packet level QoS constraints based on ILP models

MPLS (multiprotocol label switching) over WDM (wavelength division multiplexing) networks are gaining significant attention due to the efficiency in resource utilization that can be achieved by jointly considering the two network layers. This paper addresses the design of MPLS over WDM networks, where some of the WDM nodes may not have packet switching capabilities. Given the WDM network topology and the offered traffic matrix, which includes the location of the edge LSRs (label switched routers), we jointly determine the location of the core LSRs (i.e. the core WDM nodes that also need to include packet switching capabilities) and the lightpath routes (which are terminated on the LSRs) that minimize the total network cost. We consider constraints both at the optical and packet layers: an MPLS hop constraint on the maximum number of LSRs traversed by each LSP (label switched path), which guarantees a given packet level QoS, and a WDM path constraint on the maximum length of lightpaths, which accommodates the optical transmission impairments. A novel integer linear programming (ILP) formulation based on an hop-indexed approach, which we call the HOP model, is proposed. A two-phase heuristic, derived from a decomposition of the HOP model in two simpler ILP models that are solved sequentially, is also developed. The computational results show that the heuristic is efficient and produces good quality solutions, as assessed by the lower bounds computed from the HOP model. In some cases, the optimal solution is obtained with the branch-and-bound method.

Benders Decomposition for the Hop-Constrained Survivable Network Design Problem

Given a graph with nonnegative edge weights and node pairs Q, we study the problem of constructing a minimum weight set of edges so that the induced subgraph contains at least K edge-disjoint paths containing at most L edges between each pair in Q. Using the layered representation introduced by Gouveia [Gouveia, L. 1998. Using variable redefinition for computing lower bounds for minimum spanning and Steiner trees with hop constraints. INFORMS J. Comput.102 180--188], we present a formulation for the problem valid for any K, L ≥ 1. We use a Benders decomposition method to efficiently handle the large number of variables and constraints. We show that our Benders cuts contain constraints used in previous studies to formulate the problem for L = 2, 3, 4, as well as new inequalities when L ≥ 5. Whereas some recent works on Benders decomposition study the impact of the normalization constraint in the dual subproblem, we focus here on when to generate the Benders cuts. We present a thorough computational study of various branch-and-cut algorithms on a large set of instances including the real-based instances from SNDlib. Our best branch-and-cut algorithm combined with an efficient heuristic is able to solve the instances significantly faster than CPLEX 12 on the extended formulation.

On Formulations and Methods for the Hop-Constrained Minimum Spanning Tree Problem

In this chapter we present a general framework for modeling the hopconstrained minimum spanning tree problem (HMST) which includes formulations already presented in the literature. We present and survey different ways of computing a lower bound on the optimal value. These include, Lagrangian relaxation, column generation and model reformulation. We also give computational results involving instances with 40 and 80 nodes in order to compare some of the ideas discussed in the chapter.