Steiner Tree Problems With Profits

Alysson Costa, Jean-François Cordeau, Gilbert Laporte
2006 INFOR. Information systems and operational research  
This is a survey of the Steiner tree problem with profits, a variation of the classical Steiner problem where, besides the costs associated with edges, there are also revenues associated with vertices. The relationships between these costs and revenues are taken into consideration when deciding which vertices should be spanned by the solution tree. The survey contains a classification of the problems falling within this category and an overview of the methods developed to solve them. It also
more » ... ts several graph preprocessing procedures and analyzes their validity for the different variants of the problem. Finally, a brief comparison is made between the profit versions of the Steiner tree problem and of the travelling salesman problem. Let G = (V, E) be a graph with vertex set V = {1, . . . , n} and edge set E = {e = (i, j) : i, j ∈ V, i < j}, where each edge e ∈ E has an associated cost c e . The problem of determining a minimum cost network spanning all vertices V of G is known as the Minimum Spanning Tree Problem (MSTP). The Steiner Tree Problem (STP) is a very similar problem arising when some vertices need not be spanned, but may be used if their inclusion reduces the solution cost (see, for instance, Hwang, Richards and Winter, 1992) . Unlike the MSTP, the STP is NP-hard. Steiner Tree Problems with Profits (STPP) are an important variation of the classical STP. 1 In the STPP, in addition to the costs associated with the edges, there are also revenues r i , associated with the vertices i of the graph. The goal is to determine a subtree minimizing cost or maximizing revenue (or profit), subject to constraints. The exact criteria guiding the optimization vary for the different versions of the STPP. In some particular problems, both cost and revenue are combined in the objective function, while in others, limits on either the cost or the revenue will appear as constraints. Therefore, four basic criteria can be used to distinguish between the variants: the cost of edges, the revenue of vertices, the minimum collected revenue (quota), and the maximum allowed cost (budget). Depending on the problem, some of these criteria can be ignored or combined. Our aim is to review the main applications of the STPP as well as the models and algorithms proposed for its resolution. The remainder of this paper is organized as follows. Applications are described in Section 2, followed, in Section 3, by a more detailed description of the different problems and by an overview of the existing methods. An important component of modern algorithms is the application of reduction tests to eliminate some vertices and edges that are absent from at least one optimal solution. In Section 4, these preprocessing tests are reviewed and their validity is analyzed for the different variants of the STPP. Then, in Section 5, a brief comparison between the STPP and the travelling salesman problem with profits is reported. The paper ends with some conclusions in Section 6. Applications Several network design problems can be appropriately modelled as an STPP. One traditional application is the design of telecommunications local access networks. Here, the goal is to create or expand a local access network to offer service to new customers. Each new consumer represents a potential revenue for the company but there also exists a connection cost associated with the network to be constructed. The problem is clearly modelled in a graph, where customers are represented by vertices and physical links between them are represented by edges. The cost associated with an edge is the cost of laying down the optical fiber, while the revenue of a vertex is derived from customers. Ljubić et al. (2004) have reported a similar application in the planning of heating networks, where customers have an estimated heat demand and the street network provides the underlying graph where pipes can be laid down. The STPP may also appear as a subproblem of more general problems. Engevall, Göthe-Lundgren and Värbrand (1998) have solved an STPP as a subproblem within a constraint generation algorithm for the problem of cost allocation in minimum 1-trees (in a graph G, a 1-tree is a spanning tree on G = (V \{1}, E) connected to vertex 1 by two edges). Chawla et al. (2003) have used an STPP as part of a mechanism for the extended multicasting game in
doi:10.1080/03155986.2006.11732743 fatcat:uxm7hy6nujhsdhlsdb5kwl6u6a