Discrete Optimization
New formulations of the Hop-Constrained Minimum Spanning Tree problem via Miller–Tucker–Zemlin constraints

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Abstract

Given an undirected network with positive edge costs and a natural number p, the Hop-Constrained Minimum Spanning Tree problem (HMST) is the problem of finding a spanning tree with minimum total cost such that each path starting from a specified root node has no more than p hops (edges). In this paper, we develop new formulations for HMST. The formulations are based on Miller–Tucker–Zemlin (MTZ) subtour elimination constraints, MTZ-based liftings in the literature offered for HMST, and a new set of topology-enforcing constraints. We also compare the proposed models with the MTZ-based models in the literature with respect to linear programming relaxation bounds and solution times. The results indicate that the new models give considerably better bounds and solution times than their counterparts in the literature and that the new set of constraints is competitive with liftings to MTZ constraints, some of which are based on well-known, strong liftings of Desrochers and Laporte (1991).

Introduction

Minimum spanning tree problems arise quite naturally in transportation and communication network design when it is necessary to provide a minimum-cost connectivity among a number of geographically dispersed locations or system components. Various examples of minimum-cost tree networks are given by Ahuja et al. (1993) from network design in transportation, telecommunication, data storage, and cluster analysis. We consider in this paper the Hop Constrained Minimum Spanning Tree (HMST) problem, studied earlier by Gouveia (1995), which involves designing a minimum-cost spanning tree such that each path in the tree from a specified root node to every other node has no more than p hops (edges).

Gouveia (1995) has given several formulations of the HMST problem based on the well-known Miller–Tucker–Zemlin (MTZ) subtour elimination constraints (Miller et al., 1960) and has provided various liftings to MTZ constraints, some of which are based on the strong liftings of Desrochers and Laporte (1991). Linear programming (LP) bounds and Lagrangean relaxation bounds based on subgradient optimization have been offered. Reported computational results in that paper on complete graphs with up to 40 nodes indicate that lower bounds resulting from LP or Lagrangean relaxations are weak.

In this paper, we propose new formulations of the HMST problem to improve LP bounds and solution times. We propose new sets of constraints to strengthen the MTZ subtour elimination constraints as well as hop-related topology-enforcing constraints. Our computational tests indicate that the formulations we propose give considerably better LP bounds and solution times than previous formulations studied earlier by Gouveia (1995).

Hop constraints arise in designing local access networks where a central processing unit (computer) communicates with many terminals at geographically dispersed locations that are connected to it via multidrop transmission lines. A tree structure leads to a better utilization of line capacities due to shared usage. Such networks generally enjoy low message traffic. Congestion is either rare or nonexistent. While transmission time in any one link is negligible due to low traffic, traversal of many links during transmission leads to non-negligible delays that are kept under control by restricting the number of hops traversed during transmission. Hop constraints are also relevant for networks with links that are prone to possible failure. Requiring that all messages be successfully transmitted with a certain threshold probability can be expressed as a hop constraint assuming that link failures are independent identically distributed random variables. Woolston and Albin (1988) show, for example, that spanning tree designs with an upper bound on the number of hops perform better with respect to successful message transmission than those without such bounds. LeBlanc et al. (1999), Balakrishnan and Altinkemer (1992), and Gouveia et al. (2003) discuss applications of hop constraints in more general network design problems. Dahl (1998) also defines applications in transportation, statistics, and plant location for the case with at most two hops per path.

To define the HMST problem, let G = (V, E) be an undirected connected network with node set V = {r, 1, …, n}, edge set E, and positive edge costs ce (e  E). Node r represents the central processing unit and is referred to as the root node.A spanning tree of G is a connected sub-graph of G that has no cycles and spans all nodes. Given a positive integer p, a spanning tree is a hop-constrained spanning tree or a feasible tree if the unique path from the root node to any other node has no more than p hops. The HMST problem is the problem of finding a feasible tree whose total cost is minimum. If p  n, all spanning trees are feasible and hop constraints can be ignored. If p = 1, either there is no feasible tree or there is exactly one feasible tree which is the star tree with the root node at its center. We assume 2  p  n  1 from now on.

While the unconstrained minimum cost spanning tree problem is solvable in low order polynomial time by the algorithms of Kruskal (1956) and Prim (1957), the HMST problem is NP-Hard (Gouveia, 1995; Dahl, 1998). Manyem and Stallmann (1996) show that the HMST problem is not in the class APX (the class of problems for which it is possible to have a polynomial time heuristic with a guaranteed approximation bound). Dahl (1998) studies the HMST problem for p = 2 and compares the polyhedra of the models with directed and undirected arcs. Alfandari and Paschos (1999) show that the case with p = 2 cannot be approximated by polynomial time approximation schemes unless P = NP.

A version of the HMST problem that limits the number of nodes rather than edges on each path is introduced at a rudimentary level by Gavish (1985). Gouveia (1995) formulates the HMST problem using the MTZ subtour elimination constraints and studies the LP and Lagrangean relaxation bounds. An alternative formulation is offered by Gouveia (1996) based on directed or undirected multi-commodity flows (MCF). Even though the LP bounds of MCF formulations are considerably better than those of MTZ formulations, MCF formulations lead to very large integer programming models whose LP relaxations require excessive solution times and core storage as the network size and p get larger (Dahl et al., 2006). Additional improvements are obtained for small values of p using a hop-indexed formulation (Gouveia, 1998). MTZ formulations are much more compact in the numbers of variables and constraints than MCF based formulations. There are problem instances, for example, where an MTZ formulation finds an integer feasible solution in seconds whereas an MCF formulation cannot even solve the LP relaxation in days (Dahl et al., 2006). Computational results also indicate that it may take a considerable amount of time to arrive at an optimal integer solution even if the gap between the LP and integer optimal values is quite small. Due to significant differences in model sizes and in solution times between MTZ and MCF formulations, we focus in this study on the former.

Other related works on the HMST problem include lower bounding schemes based on Lagrangean relaxation (Gouveia, 1998). Reported computational results indicate that Lagrangean based bounds are much better than LP bounds of the MCF formulation for small values of p and that it is very time-consuming to obtain bounds that are close to the theoretically possible best bounds. To overcome this inefficiency, Gouveia and Requejo (2001) give a Lagrangean-based bounding scheme using a hop-indexed formulation. Even though this approach gives better bounds than the aforementioned ones, it is dependent on the value of p and performs poorly as p increases.

Dahl et al. (2004) introduce a new formulation for the HMST problem using only natural design variables and an exponential number of constraints composed of the so-called jump inequalities that are shown to be facet-defining. Their proposed formulation uses fewer variables but has weaker LP bounds than MCF formulations. Due to the exponential number of constraints, the authors propose a Lagrangean-based bounding scheme. Computational results indicate that the LP bounds are better as p increases than those reported in previous studies. Dahl et al. (2006) summarize the aforementioned approaches. Kerivin and Mahjoub (2005) give a survey of several network design problems with hop constraints and methods to solve them. More general telecommunication network design problems are also considered using MCF formulations (Balakrishnan and Altinkemer, 1992) or hop-indexed formulations (Pirkul and Soni, 2003; Gouveia et al., 2003).

The models described in the aforementioned papers view the HMST problem as defined in the original graph (although some of these models view the underlying hop-constrained shortest path problem as defined in an appropriate layered graph). Gouveia et al. (2010) propose a modeling approach that views the whole problem as defined in a single layered graph. They show that the HMST problem is equivalent to a Steiner tree problem (Hwang et al., 1992; Maculan, 1987) in an adequate layered graph. They prove that the LP bounds obtained by the directed cut Steiner tree formulation on a layered graph with an exponential number of constraints is better than the best known ones in the literature. Computational results with a branch-and-cut algorithm show that the proposed method is significantly better than previously known methods.

Gouveia et al. (2008) study a distance-constrained version of the HMST problem where edges have different transmission delays and the bound p on the number of hops is replaced by a bound on the sum of edge delays along any path from the root node. The authors present a column generation scheme, a Lagrangian relaxation procedure combined with subgradient optimization, and a shortest path (compact) reformulation which views the underlying subproblem as defined on an extended layered graph. Computational results show that the layered-graph path formulation is better than the other two approaches.

A closely related problem is the Diameter-Constrained MST problem where the aim is to find a minimum-cost spanning tree such that the unique path between any pair of nodes has no more than p hops. Some notable contributions on this problem are Achuthan et al. (1994), Gouveia and Magnanti (2003), Gruber and Raidl (2005), Santos et al. (2004), and Gouveia et al. (2010).

Our focus in the paper is on new MTZ based formulations that give better LP bounds and better solution times than the previous MTZ based formulations with or without liftings. The remainder of the paper is organized as follows. Section 2 reviews the existing MTZ based models of the HMST problem studied in Gouveia (1995). Section 3 gives our proposed formulations. Section 4 gives a description of the test problems. Section 5 compares new and existing models on the basis of LP bounds. Section 6 compares existing and new models on the basis of solution times. Section 7 concludes the paper.

Section snippets

Existing MTZ based formulations of the HMST problem

In this section, we give the formulations of the HMST problem studied by Gouveia (1995). We find it convenient to view the constraints of the minimum spanning tree problems with additional requirements on the structure of the tree in two subsets, tree-defining and topology-enforcing. This distinction is made earlier by Akgun and Tansel (2010) in the context of the Minimum-Degree Constrained Minimum Spanning Tree problem and provides a more structured way of assessing effects of different sets

New formulations for the HMST problem

The basic topology-enforcing constraints (6) establish the hop requirement by imposing the upper bound p on the node labels ui. The extensions discussed in the previous section of the basic model are obtained by liftings that make adjustments on the values of the node labels. In this sense, only one topological aspect of the problem is exploited by existing formulations. Even though not directly stated in its definition, the HMST problem has additional exploitable aspects in the topological

Test problems

Computational tests are performed using specially-structured test problems from the literature (e.g., Dahl et al., 2006). Test problems consist of three 20-node, three 40-node, and three 60-node, complete networks with 210, 820, and 1830 edges, respectively. For each network size, two Euclidean instances, TC and TE, and one random instance, TR, are considered. Euclidean instances differ from each other based on the location of the root node. In TC instances, the root node is located in the

LP bounds

We take up the discussion regarding the LP relaxation bounds in three parts. In the first and second parts, we discuss the relative standing of the existing models and proposed models among themselves, respectively. In the third part, we discuss the relative standing of the existing and new models with respect to each other. In this discussion, PL is used to represent the optimal objective function value of the LP relaxation of a linear integer programming problem P.

Table 1 gives the LP bounds

Solution times and optimality gaps

Table 3, Table 4 give solution times, optimal objective function values (if optimal solution is attained), and the best feasible objective function values together with relative optimality gaps (if optimal solution is not available) for the models offered by Gouveia (1995) and the proposed models in this paper, respectively.

The results show that all TC 20 and TR 20 instances are all solved in less than 1 second by all existing and new models. All TE 20 instances can be solved by the proposed

Tests for different hop values and network sizes

To see the effect of hop values on the LP bounds, objective function values, and solution times, the models with the best solution times, Rel-M, IMTZ/E6ITEF, and BMTZ/E7BTEF, are solved for different hop values changing from 4 to 40 by using the TE 40 instance. The results are given in Table 5.

The results show that the LP bounds and the optimal objective function values stabilize after p = 8 and p = 18, respectively, for all three models. IMTZ/E6ITEF has better LP bounds than the other two models

Conclusion

The computational results that we get in this paper indicate that the new models we propose in the paper give considerably better bounds and solution times than their counterparts in the literature and that the new sets of constraints that we propose to improve the existing formulations are competitive with earlier proposed liftings to Miller–Tucker–Zemlin constraints in Gouveia (1995), some of which are based on the well-known, strong liftings of Desrochers and Laporte (1991). Our

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