A branch and bound algorithm for the robust spanning tree problem with interval data

The robust spanning tree problem is a variation, motivated by telecommunications applications, of the classic minimum spanning tree problem. In the robust spanning tree problem edge costs lie in an interval instead of having a fixed value.Interval numbers model uncertainty about the exact cost values. A robust spanning tree is a spanning tree whose total cost minimizes the maximum deviation from the optimal spanning tree over all realizations of the edge costs. This robustness concept is formalized in mathematical terms and is used to drive optimization.In this paper a branch and bound algorithm for the robust spanning tree problem is proposed. The method embeds the extension of some results previously presented in the literature and some new elements, such as a new lower bound and some new reduction rules, all based on the exploitation of some peculiarities of the branching strategy adopted.Computational results obtained by the algorithm are presented. The technique we propose is up to 210 faster than methods recently appeared in the literature.     

The complexity of a minimum reload cost diameter problem

We consider the minimum diameter spanning tree problem under the reload cost model which has been introduced by Wirth and Steffan [H.-C. Wirth, J. Steffan, Reload cost problems: Minimum diameter spanning tree, Discrete Appl. Math. 113 (2001) 73-85]. In this model an undirected edge-coloured graph G is given, together with a nonnegative symmetrical integer matrix R specifying the costs of changing from a colour to another one. The reload cost of a path in G arises at its internal nodes, when passing from the colour of one incident edge to the colour of the other. We prove that, unless P=NP, the problem of finding a spanning tree of G having a minimum diameter with respect to reload costs, when restricted to graphs with maximum degree 4, cannot be approximated within any constant α<2 if the reload costs are unrestricted, and cannot be approximated within any constant β<5/3 if the reload costs satisfy the triangle inequality. This solves a problem left open by Wirth and Steffan [H.-C. Wirth, J. Steffan, Reload cost problems: minimum diameter spanning tree, Discrete Appl. Math. 113 (2001) 73-85].     

A Benders decomposition approach for the robust spanning tree problem with interval data

The robust spanning tree problem is a variation, motivated by telecommunications applications, of the classic minimum spanning tree problem. In the robust spanning tree problem edge costs lie in an interval instead of having a fixed value.Interval numbers model uncertainty about the exact cost values. A robust spanning tree is a spanning tree whose total cost minimizes the maximum deviation from the optimal spanning tree over all realizations of the edge costs. This robustness concept is formalized in mathematical terms and is used to drive optimization.This paper describes a new exact method, based on Benders decomposition, for the robust spanning tree problem with interval data. Computational results highlight the efficiency of the new method, which is shown to be very fast on all the benchmarks considered, and in particular on those that were harder to solve for the methods previously known.     

Solving the generalized minimum spanning tree problem by a branch-and-bound algorithm

We present an exact algorithm for solving the generalized minimum spanning tree problem (GMST). Given an undirected connected graph and a partition of the graph vertices, this problem requires finding a least-cost subgraph spanning at least one vertex out of every subset. In this paper, the GMST is formulated as a minimum spanning tree problem with side constraints and solved exactly by a branch-and-bound algorithm. Lower bounds are derived by relaxing, in a Lagrangian fashion, complicating constraints to yield a modified minimum cost spanning tree problem. An efficient preprocessing algorithm is implemented to reduce the size of the problem. Computational tests on a large set of randomly generated instances with as many as 250 vertices, 1000 edges, and 25 subsets provide evidence that the proposed solution approach is very effective.     

A multi-population hybrid biased random key genetic algorithm for hop-constrained trees in nonlinear cost flow networks

Genetic algorithms and other evolutionary algorithms have been successfully applied to solve constrained minimum spanning tree problems in a variety of communication network design problems. In this paper, we enlarge the application of these types of algorithms by presenting a multi-population hybrid genetic algorithm to another communication design problem. This new problem is modeled through a hop-constrained minimum spanning tree also exhibiting the characteristic of flows. All nodes, except for the root node, have a nonnegative flow requirement. In addition to the fixed charge costs, nonlinear flow dependent costs are also considered. This problem is an extension of the well know NP-hard hop-constrained Minimum Spanning Tree problem and we have termed it hop-constrained minimum cost flow spanning tree problem. The efficiency and effectiveness of the proposed method can be seen from the computational results reported.     

A composite very large-scale neighborhood structure for the capacitated minimum spanning tree problem

The capacitated minimum spanning tree (CMST) problem is to find a minimum cost spanning tree in a network where nodes have specified demands, with an additional capacity constraints on the subtrees incident to a given source node s. The capacitated minimum spanning tree problem arises as an important subproblem in many telecommunication network design problems. In a recent paper, Ahuja et al. (Math. Program. 91 (2001) 71) proposed two very large-scale neighborhood search algorithms for the capacitated minimum spanning tree problem. Their first node-based neighborhood structure is obtained by performing multi-exchanges involving several trees where each tree contributes a single node. Their second tree-based neighborhood structure is obtained by performing multi-exchanges where each tree contributes a subtree. The computational investigations found that node-based multi-exchange neighborhood gives the best performance for the homogenous demand case (when all nodes have the same demand), and the tree-based multi-exchange neighborhood gives the best performance for the heterogeneous demand case (when nodes may have different demands). In this paper, we propose a composite neighborhood structure that subsumes both the node-based and tree-based neighborhoods, and outperforms both the previous neighborhood search algorithms for solving the capacitated minimum spanning tree problem on standard benchmark instances. We also develop improved dynamic programming based exact algorithms for searching the composite neighborhood.     

Efficient algorithms for a family of matroid intersection problems

Consider a matroid where each element has a real-valued cost and a color, red or green; a base is sought that contains q red elements and has smallest possible cost. An algorithm for the problem on general matroids is presented, along with a number of variations. Its efficiency is demonstrated by implementations on specific matroids. In all cases but one, the running time matches the best-known algorithm for the problem without the red element constraint: On graphic matroids, a smallest spanning tree with q red edges can be found in time O(n log n) more than what is needed to find a minimum spanning tree. A special case is finding a smallest spanning tree with a degree constraint; here the time is only O(m + n) more than that needed to find one minimum spanning tree. On transversal and matching matroids, the time is the same as the best-known algorithms for a minimum cost base. This also holds for transversal matroids for convex graphs, which model a scheduling problem on unit-length jobs with release times and deadlines. On partition matroids, a linear-time algorithm is presented. Finally an algorithm related to our general approach finds a smallest spanning tree on a directed graph, where the given root has a degree constraint. Again the time matches the best-known algorithm for the problem without the red element (i.e., degree) constraint.     

“Optimistic” weighted Shapley rules in minimum cost spanning tree problems

We introduce optimistic weighted Shapley rules in minimum cost spanning tree problems. We define them as the weighted Shapley values of the optimistic game v⁺ introduced in Bergantiños and Vidal-Puga [Bergantiños, G., Vidal-Puga, J.J., forthcoming. The optimistic TU game in minimum cost spanning tree problems. International Journal of Game Theory. Available from: <http://webs.uvigo.es/gbergant/papers/cstShapley.pdf>]. We prove that they are obligation rules [Tijs, S., Branzei, R., Moretti, S., Norde, H., 2006. Obligation rules for minimum cost spanning tree situations and their monotonicity properties. European Journal of Operational Research 175, 121–134].     

A vertex oriented approach to the equal remaining obligations rule for minimum cost spanning tree situations

In this paper, we consider spanning tree situations, where players want to be connected to a source as cheap as possible. These situations involve the construction of a spanning tree with the minimum cost as well as the allocation of the cost of this minimum cost spanning tree among its users in a fair way. Feltkamp, Muto and Tijs 1994 introduced the equal remaining obligations rule to solve the cost allocation problem in these situations. Recently, it has been shown that the equal remaining obligations rule satisfies many appealing properties and can be obtained with different approaches. In this paper, we provide a new approach to obtain the equal remaining obligations rule. Specifically, we show that the equal remaining obligations rule can be obtained as the average of the cost allocations provided by a vertex oriented construct-and-charge procedure for each order of players.     

Transitions in geometric minimum spanning trees

We study some combinatorial and algorithmic problems associated with an arbitrary motion of input points in space. The motivation for such an investigation comes from two different sources:computer modeling andsensitivity analysis. In modeling, the dynamics enters the picture since geometric objects often model physical entities whose positions can change over time. In sensitivity analysis, the motion of the input points might represent uncertainties in the precise location of objects. The main results of the paper deal with state transitions in the minimum spanning tree when one or more of the input points move arbitrarily in space. In particular, questions of the following form are addressed: (i) How many different minimum spanning trees can arise if one point moves while the others remain fixed? (ii) When does the minimum spanning tree change its topology if all points are allowed to move arbitrarily?     

Obligation rules for minimum cost spanning tree situations and their monotonicity properties

We consider the class of Obligation rules for minimum cost spanning tree situations. The main result of this paper is that such rules are cost monotonic and induce also population monotonic allocation schemes. Another characteristic of Obligation rules is that they assign to a minimum cost spanning tree situation a vector of cost contributions which can be obtained as product of a double stochastic matrix with the cost vector of edges in the optimal tree provided by the Kruskal algorithm. It turns out that the Potters value (P-value) is an element of this class.     

On the complexity of some arborescences finding problems on a multihop radio network

An arborescence of a multihop radio network is a directed spanning tree (with rootx) such that the edges are directed away from the root. Based upon an arborescence,x canbroadcast a message to other nodes according to the directed edges of the spanning tree. The minimum transmission power arborescence problem is to find an arborescence such that the message can be broadcasted to other nodes by using a minimal amount of transmission power. The minimum delay arborescence problem is to find an arborescence such that a message can be broadcasted to other nodes by using a minimal number of broadcast transmission. In this paper we show that both these problems areNP-complete. The reductions are from the maximum leaf spanning tree problem.Areverse arborescence is similar to an arborescence except that the edges are directed toward the root. Based upon a reverse arborescence, the root node cancollect information from other nodes. In this paper we also show that the reverse minimum transmission power arborescence problem can be solved with the same computational complexity as that of finding a minimum cost spanning tree, and the reverse minimum delay arborescence problem can be solved with the same computational complexity as that of finding a spanning tree.     

Spanning Trees Crossing Few Barriers

We consider the problem of finding low-cost spanning trees for sets of $n$ points in the plane, where the cost of a spanning tree is defined as the total number of intersections of tree edges with a given set of $m$ barriers. We obtain the following results: (i) if the barriers are possibly intersecting line segments, then there is always a spanning tree of cost $O(\min(m^2,m\sqrt{n}))$; (ii) if the barriers are disjoint line segments, then there is always a spanning tree of cost $O(m)$; (iii) ] if the barriers are disjoint convex objects, then there is always a spanning tree of cost $O(n+m)$. All our bounds are worst-case optimal, up to multiplicative constants.     

On the inverse problem of minimum spanning tree with partition constraints

In this paper we first discuss the properties of minimum spanning tree and minimum spanning tree with partition constraints. We then concentrate on the inverse problem of minimum spanning tree with partition constraints in which we need to adjust the weights of the edges in a network as less as possible so that a given spanning tree becomes the minimum one among all spanning trees that satisfy the partition restriction. Based on the calculation of maximum cost flow in networks, we propose a strongly polynomial algorithm for solving the problem.The author gratefully acknowledges the partial support of Croucher Foundation.     

Opportune moment strategies for a cost spanning tree game

Cost spanning tree problems concern the construction of a tree which provides a connection with the source for every node of the network. In this paper, we address cost sharing problems associated to these situations when the agents located at the nodes act in a non-cooperative way. A class of strategies is proposed which produce minimum cost spanning trees and, at the same time, are strong Nash equilibria for a non-cooperative game associated to the problem. They are also subgame perfect Nash equilibria.     

Minimum cost spanning tree games

We consider the problem of cost allocation among users of a minimum cost spanning tree network. It is formulated as a cooperative game in characteristic function form, referred to as a minimum cost spanning tree (m.c.s.t.) game. We show that the core of a m.c.s.t. game is never empty. In fact, a point in the core can be read directly from any minimum cost spanning tree graph associated with the problem. For m.c.s.t. games with efficient coalition structures we define and construct m.c.s.t. games on the components of the structure. We show that the core and the nucleolus of the original game are the cartesian products of the cores and the nucleoli, respectively, of the induced games on the components of the efficient coalition structure.This paper is a revision of [4].     

Chance constrained bottleneck spanning tree problem

In this paper we consider a stochastic version of the bottleneck spanning tree in which edge costs are independent random variables. Our problem is to find an optimal spanning tree and optimal satisficing level of the chance constraint with respect to the bottleneck (maximum cost) edge of the spanning tree. The problem is first transformed into a deterministic equivalent problem. Then a subproblem in order to solve the problem is introduced and the close relation between these problems is clarified. Finally, based on the relation, polynomial time solution procedures to solve the problem are proposed under two special functions of satisficing level which are given as examples to be solved easily.     

On the optimal management of project risk

The uncertainty of project networks has been mainly considered as the randomness of duration of the activities. However, another major problem for project managers is the uncertainty due to the randomness of the amount of resources required by each activity which can be expressed by the randomness of its cost. Such randomness can seriously affect the discounted cost of the project and it may be strongly correlated with the duration of the activity.In this paper, a model considering the randomness of both the cost and the duration of each activity is introduced and the problem of project scheduling is studied in terms of the project's discounted cost and of the risk of not meeting its completion time. The adoption of the earliest (latest) starting time for each activity decreases (increases) the risk of delays but increases (decreases) the discounted cost of the project. Therefore, an optimal compromise has to be achieved. This problem of optimization is studied in terms of the probability of the duration and of the discounted cost of the project falling outside the acceptable domain (Risk function) using the concept of float factor as major decision variable. This last concept is proposed to help the manager to synthetize the large number of the decision variables representing each schedule for the studied project. Numerical results are also presented for a specific project network.     

On generalized minimum spanning trees

The generalized minimum spanning tree problem consists of designing a minimum cost tree spanning several clusters. The purpose of this note is to pinpoint several inaccuracies contained in a previous publication and to propose a valid formulation for this problem.     

Connection problems in mountains and monotonic allocation schemes

Directed minimum cost spanning tree problems of a special kind are studied, namely those which show up in considering the problem of connecting units (houses) in mountains with a purifier. For such problems an easy method is described to obtain a minimum cost spanning tree. The related cost sharing problem is tackled by considering the corresponding cooperative cost game with the units as players and also the related connection games, for each unit one. The cores of the connection games have a simple structure and each core element can be extended to a population monotonic allocation scheme (pmas) and also to a bi-monotonic allocation scheme. These pmas-es for the connection games result in pmas-es for the cost game.