Network synthesis problem : cost allocation and algorithms
Creators: name: Hojati, Mehran
Publisher: name: University of British Columbia
Year: 2010

description: This thesis is concerned with a network design problem which is referred to in the literature as the network synthesis problem. The objective is to design an undirected network, at a minimum cost, which satisfies known requirements, i.e., lower bounds on the maximum flows, between every pair of nodes. If the requirements are to be satisfied nonsimultaneously, i.e., one at a time, the problem is called the nonsimultaneous network synthesis problem, whereas if the requirements are to be satisfied simultaneously, the problem is called the simultaneous network synthesis problem. The total construction cost of the network is the sum of the construction cost of capacities on the edges, where the construction cost of a unit capacity is fixed for any edge, independent of the size of the capacity, but it may differ from edge to edge. The capacities are allowed to assume noninteger nonnegative values. The simultaneous network synthesis problem was efficiently solved by Gomory and Hu [60], whereas the nonsimultaneous network synthesis problem can only be formulated and solved as a linear program with a large number of constraints. However, the special equal-cost case, i.e., when the unit construction costs are equal across the edges, can be efficiently solved, see Gomory and Hu [60], by some combinatorial method, other than linear programming. A cost allocation problem which is associated with the network synthesis problem would naturally arise, if we assume that the various nodes in the network represent different users or communities. In this case, we need to find a fair method for allocating the construction cost of the network among the different users. An interesting generalization of the nonsimultaneous network synthesis problem, the Steiner network synthesis problem, is derived, when only a proper subset of the nodes have positive requirements from each other. The thesis is concerned with two issues. First, we will analyze the cost allocation problems arising in the simultaneous and the equal cost nonsimultaneous network synthesis problem. Secondly, we will consider the Steiner network synthesis problem, with particular emphasis on simplifying the computations in some special cases, not considered before. We will employ cooperative game theory to formulate the cost allocation problems, and we will prove that the derived games are 'concave', which implies the existence of the core and the inclusion of the Shapley value and the nucleolus in the core, and then we will present irredundant representations of the cores. For the equal cost nonsimultaneous network synthesis game, we will use the irredundant representation of the core to provide an explicit closed form expression for the nucleolus of the game, when the requirement structure is a spanning tree; then, we will develop, in a special case, a decomposition of the game, which we will later use to efficiently compute the Shapley value of the game when the requirement structure is a tree; the decomposition will also be used for the core and the nucleolus of the game in the special case. For the simultaneous network synthesis game, we will also use the irredundant representation of the core to derive an explicit closed form expression for the nucleolus, and we will also decompose the core of this game in the special case, and prove that the Shapley value and the nucleolus coincide. Secondly, for the Steiner network synthesis problem, two conceptually different contributions have been made, one being a simplifying transformation, and the other being the case when the network has to be embedded in (i.e., restricted to) some special graphs. Namely, when the requirement structure is sparse (because there are only a few demand nodes and the rest are just intermediate nodes) and the positive requirements are equal, we will employ a transformation procedure to simplify the computations. This will enable us to efficiently solve the Steiner network synthesis problem with five or less nodes which have equal positive requirements from each other. Finally, when the solution network to the Steiner network synthesis problem is to be embedded in (restricted to) some special graphs, namely trees, rings (circles), series-parallel graphs, or M₂ and M₃-free graphs, we will provide combinatorial algorithms which are expected to simplify the computations.