Computational Complexity of a Cost Allocation Approach to a Fixed Cost Spanning Forest Problem

We present a computational analysis of a game theoretic approach to a cost allocation problem arising from a graph optimization problem, referred to as the fixed cost spanning forest FCSF problem. The customers in the FCSF problem, represented by nodes in a graph G, are in need of service that can be produced at some facilities yet to be constructed. The cost allocation problem is concerned with the fair distribution of the cost of providing the service among customers. We formulate this cost allocation problem as a cooperative game, referred to as the FCSF game. In general, the core of a FCSF game may be empty. However, for the case when G is a tree, it is shown that the core is not empty. Moreover, we prove that in this case core points can be generated in strongly polynomial time. We further provide a nonredundant characterization of the core of the FCSF game defined over a tree in the special case when all nodes are communities. This is shown to lead, in some instances, to a strongly polynomial algorithm for computing the nucleolus.

[1]  A. Kopelowitz.  COMPUTATION OF THE KERNELS OF SIMPLE GAMES AND THE NUCLEOLUS OF N-PERSON GAMES. , 1967 .

[2]  C. G. Bird,et al.  On cost allocation for a spanning tree: A game theoretic approach , 1976, Networks.

[3]  A. Charnes,et al.  ON A CLASS OF CONVEX AND NON-ARCHIMEDEAN SOLUTION CONCEPTS FOR N-PERSON GAMES. , 1967 .

[4]  Nimrod Megiddo,et al.  Computational Complexity of the Game Theory Approach to Cost Allocation for a Tree , 1978, Math. Oper. Res..

[5]  L. Shapley.  A Value for n-person Games , 1988 .

[6]  L. Shapley.  Cores of convex games , 1971 .

[7]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[8]  Éva Tardos,et al.  A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs , 1986, Oper. Res..

[9]  Charles J. Colbourn,et al.  Steiner trees, partial 2-trees, and minimum IFI networks , 1983, Networks.

[10]  Nimrod Megiddo,et al.  Cost allocation for steiner trees , 1978, Networks.

[11]  D. Schmeidler.  The Nucleolus of a Characteristic Function Game , 1969 .

[12]  Richard T. Wong,et al.  A dual ascent approach for steiner tree problems on a directed graph , 1984, Math. Program..

[13]  S. Littlechild.  A simple expression for the nucleolus in a special case , 1974 .

[14]  Guillermo Owen,et al.  On the core of linear production games , 1975, Math. Program..

[15]  Morton D. Davis,et al.  The kernel of a cooperative game , 1965 .

[16]  G. Owen,et al.  A Simple Expression for the Shapley Value in a Special Case , 1973 .

[17]  Daniel Granot,et al.  A generalized linear production model: A unifying model , 1986, Math. Program..

[18]  L. S. Shapley,et al.  Geometric Properties of the Kernel, Nucleolus, and Related Solution Concepts , 1979, Math. Oper. Res..

[19]  Arie Tamir,et al.  On the core of network synthesis games , 1991, Math. Program..

[20]  L. Shapley,et al.  The kernel and bargaining set for convex games , 1971 .