Arboricity Games: the Core and the Nucleolus
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The arboricity of a network is the minimum number of forests needed to cover all edges of the network, which measures the network density [24]. We study the arboricity from a game theoretic perspective and consider cost allocations in the minimum forest cover problem. We introduce the arboricity game as a cooperative cost game defined on a graph G = (V, E), where the player set is E and the cost of a coalition S ⊆ E is the arboricity of the subgraph induced by S. We study properties of the core and propose an efficient algorithm for computing the nucleolus when the core is nonempty. To compute the nucleolus, we introduce a graph decomposition based on the density, namely the prime partition, which may be of independent interest. The prime partition decomposes the edge set of a graph into a non-prime set and a number of prime sets. The non-prime set is the set of edges that are not in any densest subgraph. The prime sets are the incremental edge sets of a chain of densest subgraphs under inclusion. For the arboricity game defined on the same graph, a partial order can be defined on the prime sets according to core allocations, and then the nucleolus can be computed efficiently by solving only two linear programs in the Maschler scheme [21, 22]. In addition to computing the nucleolus of arboricity games, the prime partition provides an analogous graph decomposition to [29, 30], which complements another line of research.
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