Darko Skorin-Kapov

Tight linear programming relaxations of uncapacitated p-hub median problems

Abstract The problem of locating hub facilities and allocating non-hub nodes to those hubs arises frequently in the design of communication networks, airline passenger flow and parcel delivery networks. In this paper we consider uncapacitated multiple and single allocation p -hub median problems. We develop new mixed 0 1 linear formulations with tight linear programming relaxations. The approach is tested on a well known and heavily used benchmark data set of real-world problems with resulting LP relaxations ranging from 10010 to 391 250 variables and from 2 101 to 31 901 constraints, which proved to be difficult linear programs. Yet, this approach proved to be very effective: in almost all instances the linear programming solution was integer. In cases with fractional solutions, the integrality was achieved by adding a small partial set of integrality constraints. Therefore, we extended the range of optimally solvable instances of these NP-hard hub location problems, which have defied researchers for the last ten years. As an additional result for the single allocation case we were able to establish optimality of all heuristic solutions obtained via tabu search algorithm from a previous study. For the more difficult single allocation p -hub median problem we also used the best known heuristic solution as a guidance in adding integrality constraints. This novel linkage between optimal and heuristic solutions has a potential impact in a number of other problem settings, where efficient heuristic solutions exist and are probably, but not provably optimal.

On tabu search for the location of interacting hub facilities

Abstract A new heuristic method based on tabu search is developed for the problem of locating p interacting hub facilities among n interacting nodes in a network. The method treats equally the problem of locating hub facilities, as well as the problem of allocating the nodes to one and only one hub. The algorithm obtained improved solutions to problems from the standard test set from literature which has been used in this study.

HUB NETWORK DESIGN WITH SINGLE AND MULTIPLE ALLOCATION: A COMPUTATIONAL STUDY

Abstract We present exact solutions for hub location models and discuss sensitivity of these solutions to the interhub discount factor. Both multiple and single hub allocations are considered. We employ a linearization that is smaller than any previously used, and we determine the number of variables and constraints. This formulation is used to make extensive computations. Specifically, we include runs for a standard test problem for 3 and 4 hubs, and n = 10, 15, 20 and 25 origins and destinations, although the case with 25 origins and destinations is still large enough to require us to present results for fewer parameter combinations than for the other cases. The results show that the integer-friendliness of the formulation depends on the value of α. A new result in this paper is a determination of the optimal number of hubs as the fixed costs and interhub discount factors change.

Network cost minimization using threshold-based discounting

Abstract A network design problem in which every pair of nodes can communicate directly is discussed. However, there is an incentive to combine flow from different sources, namely, if the total flow through a link exceeds the prescribed threshold, then the cost of this flow is discounted by a factor α. Alternative mixed integer linear formulations for this problem are presented. Computational results comparing the models on a set of benchmark problems are also presented. The results show the effectiveness of the formulations: for discounts of 5–10%, the gaps between linear and integer solutions are within few percent. Such a model offers economic incentives in building and utilizing communication networks.

On minimum cost isolated failure immune networks

A telecommunications network is isolated failure immune (IFI) if and only if communication between operative sites can be completed as long as network failures are isolated. It is known that the class of minimal IFI networks is equivalent to the class of spanning 2-trees. To the best of our knowledge, this work is the first computational study dealing with the construction of a minimum cost IFI network. The problem is known to be NP-complete. We develop a tabu search based heuristic for solving the minimum cost spanning 2-tree (MCS2T) problem. The complex structure of 2-trees makes the tabu search heuristic highly dependent on the starting solution. We develop four heuristic algorithms to obtain diversified “good” starting solutions. They are: completion of a 2-tree from a spanning tree, two greedy approaches, and a method based on the recursive definition of a 2-tree. We also formulate an integer programming problem (IP) whose objective function value is a lower bound to the MCS2T problem. We solve the IP by developing a constraint generation scheme. The algorithms were tested on complete random graphs with Euclidean distances and on two real data sets (Civil Aeronautics Board) with instances of 10, 15, 20 and 25 nodes. As a result of this research for “small” problems (10 and 15 nodes), the heuristic solutions are on average within 0.8% from the optimal solution and for “large” problems (20 and 25 nodes), the average error is less than 2.8%.

On the core of the minimum cost steiner tree game in networks

A cost allocation problem arising from the Steiner Tree (ST) problem in networks is analyzed. This cost allocation problem is formulated as a cost cooperative game in characteristic function form, referred to as theST-game. The class ofST games generalizes the class of minimum cost spanning tree games which were used in the literature to analyze a variety of cost allocation problems. In general, the core of anST-game may be empty. We construct an efficient Core Heuristic to compute a “good” lower bound on the maximum fraction of the total cost that can be distributed among users while satisfying the core constraints. Based on the Core Heuristic, we also provide a sufficient condition for a givenST not to be optimal for the linear programming relaxation of an integer programming formulation of theST problem. The Core Heuristic was implemented and tested on 76 data sets from the literature (Wongs, Anejas and Beasleys Steiner tree problems). Core points were found for 69 of these cases, and points “close” to the core were computed in the others.

On some optimization problems on k -trees and partial k -trees

A k-tree is a graph that can be reduced to the k-complete graph by sequentially removing k-degree vertices with completely connected neighbors. Partial k-trees are graphs embeddable in a k-tree with the same vertex set. In this paper we develop efficient algorithms for several path distance optimization problems on partial k-trees, and k-cable distance optimization problems on k-trees. Specifically, we develop a linear time algorithm to find shortest simple paths from a given vertex to all other vertices in a partial k-tree, we compute the diameter of a partial k-tree with equal edge lengths in linear time, and we construct an O(nk + 2) algorithm to solve the simple plant location problem in an n-vertex partial k-tree. Then, we analyze some cable distance optimization problems in k-trees. We derive some properties of cable distance in k-trees and we present a new characterization of a k-path in k-trees. Finally, we develop algorithms to solve a certain k-cable decomposition problem in k-trees in O(n2) time and to compute the k-cable diameter of a k-tree with equal edge lengths in linear time.

An efficient characterization of some cost allocation solutions associated with capacitated network design problems

We analyze some game-theoretic solution concepts associated with a cost allocation problem arising from the Capacitated Network Design (CND) problem. The problem is formulated as a cost cooperative game in characteristic function form to be referred to as the CND game. We provide an efficient representation of several game-theoretic solution concepts associated with the CND game. In particular, we efficiently characterize the core, and in some cases the nucleolus, the least weightedε-core and a certain “central” point in the least weightedε-core. Our model properly generalizes several previously studied cooperative games. We also employ our model to analyze cost allocation problems associated with several classes of network design problems, which were not previously studied in the literature. Specifically, we efficiently characterize the above cost allocation solutions for cost allocation problems associated with the Capacitated Concentrator Location problem, the Capacitated Minimum Spanning Tree problem, the Capacitated Fixed Cost Spanning Forest problem, and the Capacitated Steiner Tree problem.

NC Algorithms for Recognizing Partial 2-Trees and 3-Trees

The existence of a k-separator in a partial k-tree graph is proved and a linear time algorithm is constructed that finds such a separator in k-trees. This algorithm can be used to obtain a balanced binary decomposition of a k-tree in

A note on Steiner tree games

O( n\log n )