Network Connectivity Game

We investigate the cost allocation strategy associated with the problem of providing service /communication between all pairs of network nodes. There is a cost associated with each link and the communication between any pair of nodes can be delivered via paths connecting those nodes. The example of a cost efficient solution which could provide service for all node pairs is a (non-rooted) minimum cost spanning tree. The cost of such a solution should be distributed among users who might have conflicting interests. The objective of this paper is to formulate the above cost allocation problem as a cooperative game, to be referred to as a Network Connectivity (NC) game, and develop a stable and efficient cost allocation scheme. The NC game is related to the Minimum Cost Spanning Tree games and to the Shortest Path games. The profound difference is that in those games the service is delivered from some common source node to the rest of the network, while in the NC game there is no source and the service is established through the two-way interaction among all pairs of participating nodes. We formulate Network Connectivity (NC) game and construct an efficient cost allocation algorithm which finds some points in the core of the NC game. Finally, we discuss the Egalitarian Network Cost Allocation (ENCA) rule and demonstrate that it finds an additional core point

On Cost Allocation in Networks with Threshold Based Discounting

We study network design in which each pair of nodes can communicate via a direct link and the communication flow can be delivered through any path in the network. The cost of flow through each link is discounted if and only if the amount of flow exceeds certain threshold. This exploitation of economies of scale encourages the concentration of flows and use of relatively small number of links. Applications include telecommunications, airline traffic flow, and mail delivery networks. The cost of services delivered through such a network is distributed among its users who may be individuals or organizations with possibly conflicting interests. The cooperation between these users is essential for the exploitation of economies of scale. Consequently, there is a need to ensure a fair distribution of the cost of providing the service among network users. In order to describe this cost allocation problem we formulate the associated cooperative game, to be referred to as the threshold game. We then demonstrate that certain cost allocation solution (the core of the threshold game) can be efficiently applied to relatively ’large’ networks with threshold-based discounting

On Delay versus Congestion in Designing Rearrangeable Multihop Lightwave Networks

We investigate design issues of optical networks in light of two conflicting criteria: throughput maximization (or, equivalently, congestion minimization) versus delay minimization. We assume the network has an arbitrary topology, the flow can be split and sent via different routes, and it can be transferred via intermediate nodes. Tabu search heuristic is used to compare solutions with different weights assigned to each of the two criteria. The approach is tested on a benchmark data set, the 14-dimensional NSFNET T1 network with traffic from 1993. The results suggest that (1) some connectivity matrices are quite robust and desirable regarding both criteria simultaneously; (2) forcing minimization of total delay unconditionally can result with significantly inferior throughput. Some decisions strategies are outlined

The Monotonic Cost Allocation Rule in Steiner Tree Network Games

We investigate the cost allocation strategy associated with the problem of providing some network service from source to a number of users, via the Minimum Cost Steiner Tree Network that spans the source and all the receivers. The cost of such a Steiner tree network, is distributed among its receivers. The objective of this paper is to develop a reasonably fair and computationally efficient cost allocation rule associated with the above cost allocation problem. Since finding the optimal Steiner tree is an NP-hard problem, the input to our cost allocation problem is the best known solution obtained using some heuristic. In order to allocate the cost of this Steiner tree to the users (receiver nodes), we formulate the associated Steiner Tree Network (STN) game in characteristic function form. It is well known that the core of the general STN game might be empty. We propose a new cost allocation rule for the modified STN game which might be attractive to network users due to its monotonic properties, associated with network growth

On a Cost Allocation Problem Arising from a Star-Star Capacitated Concentrator Location Problem

We analyze a cost allocation problem associated with the Star-Star Capacitated Concentrator Location (SSCCL) problem. The problem is formulated as a cost cooperative game in characteristic function form to be referred to as the SSCCL game. The characterization and computation of game theoretic solution concepts associated with this game are investigated. We show that, in general, the core of this cooperative game may be empty. However, we provide a polynomial representation of the core of the SSCCL game. In case of nonemptiness of the core we provide an efficient method to find the nucleolus. For the case when the core is empty, we propose the least weighted e-core as a concept for fair cost allocation for the SSCCL problem and give its polynomial characterization. Moreover, certain \u27central\u27 point of the least weighted e-core is also efficiently characterized

On Hub Location Models

The study of hub location models involves designing communication networks where some of the nodes serve as focal points (i.e. hubs) and other nodes are connected to those hubs. Possible applications include airline traffic flow, telecommunications, and mail delivery networks. In this paper we present an overview of recent results on solvability of some hub location models. The overview includes a heuristic approach based on tabu search, lower bounds for cases where distances satisfy triangular inequality, tight linear programming relaxations, and a linkage between optimal and heuristic solutions. As a result of those studies the range of optimally solvable instances of NP-hard hub location problems was extended. In particular, well known and heavily used bench-mark data set of real world problems (Civil Aeronautics Board (CAB) data set), that has resisted efficient solutions for more than a decade, has been solved to optimality. The paper concludes with the discussion of some avenues for future research

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

The objective of this thesis is to investigate some structural and algorithmic properties of k-trees and partial k-trees. A k-tree can be constructed from a k-complete graph by recursively adding a new vertex which is adjacent to all vertices of an existing k-complete subgraph. Partial k-trees are graphs embeddable in a k-tree with the same vertex set. They are natural generalizations of forests and series-parallel graphs which are the first two members of the hierarchy of partial k-trees. The many applications of k-trees and partial k-trees have motivated their study from both an algorithmic and a theoretical point of view. For example, k-trees arise in reliable communication network design problems (Farley (1981), Farley and Proskurowski (1982), Neufeld and Colbourn (1983), Wald and Colbourn (1983), Colbourn and Proskurowski (1984)) and in the study of the complexity of certain type of queries in a relational data base system (Arnborg (1979)). Moreover, the class of k-trees is special in the sense that many problems, which are NP-complete for arbitrary graphs, are solvable in polynomial time when restricted to k-trees or partial k-trees (Arnborg and Proskurowski (1989)). In Chapter 2 of the thesis we analyze a fixed cost spanning forest (FCSF) problem, defined over a graph G, in which some customers require service that can be generated at some facilities' sites. Both the set of customers and facilities' sites are represented by nodes in G. There is a fixed cost for opening each facility and a cost for delivering the service from open facilities to the customers. Customers do not necessarily have to receive the service directly from an open facility, but possibly through other intermediate customers. We develop a linear time algorithm for solving the FCSF problem when the customers and potential facilities' sites are located on a series-parallel network or, equivalently, a partial 2-tree. We further analyze a related cost allocation problem, in which we seek a fair method for allocating the cost of providing the service to the customers. We formulate this cost allocation problem as a cooperative game and show that, in general, the core of this cooperative game may be empty. However, we provide a sufficient condition, which can be verified in polynomial time, for the nonemptiness of the core of this game. A k-tree can be reduced to the k-complete graph by sequentially removing k-degree vertices with completely connected neighbors. We use this reduction process to develop, in Chapter 3, efficient algorithms for several optimization problems on k-trees and partial k-trees. In particular, we develop a linear time algorithm to find shortest simple paths from a given vertex to all other vertices in a k-tree, we compute the diameter of a k-tree with equal edge lengths in linear time, and we construct an O(n[sup k+2]) algorithm to solve the Simple Plant Location problem in an n-vertex partial k-tree. In Chapter 4 we present a new characterization of a k-path between two vertices u and v, in an equal weight k-tree G, by means of minimal k and k+1 cliques with respect to certain partial orders defined on the collections of all k and k+1 cliques in a k-tree. We use it to develop an O(n²) algorithm to decompose a vertex set V of a k-tree G to a minimum number of components, such that for any pair of vertices i and j in the same component, the cable distance between i and j is bounded by a positive integer R. We also compute the k-cable diameter of a k-tree with equal edge lengths in linear time. In Chapter 5 we derive some separation properties of partial k-trees and use them to develop NC algorithms for recognizing partial 2-trees and 3-trees. Explicitly, we prove the existence of a k-separator in a partial k-tree graph and construct a linear time algorithm that finds such a separator in k-trees. This algorithm can be used to obtain a balanced binary decomposition of a k-tree in 0(n log n) time. We derive some other separation properties of partial k-trees and use them to construct a balanced decomposition of an embedding of a k-connected partial k-tree when k = 2 and 3. Finally, we construct NC algorithms for the recognition of a partial k-tree for k - 2 and 3, which run in O(log²n) time using, respectively, O(n³) and O(n⁴) processors.Business, Sauder School ofGraduat