Restringindo o espaço de busca na geração de estruturas de coalizão utilizando grafos - Nunes, Anderson Afonso (2015)
The coalition structures generating problem (CSG) involves partitioning the set of agents in all possible subsets (or coalitions). What makes this problem challenging is the number of possible coalitions that grows exponentially as new agents are inserted. The number of coalitions is (2n − 1) where n is the number of agents. However, in many real-world applications, there are inherent limitations on possible coalitions: for example, some individuals may be prohibited from being in the same coalition or coalition structure may be required to contain coalitions of the same size. When we consider CSG restricted by graphs where the viability of a coalition is restricted by a synergy graph, the computational complexity can be maintained or eventually be smaller depending on what is considered a valid coalition. Synergy graphs are representations of the agents as being the vertices and their relationships are the edges. This work is a study on the use of restrictions involving graphs as a heuristic about coalitions for the problem coalition enumeration in order to consider a feasible coalition or not according to the density of the subgraph induced by the agents. Current works using the restriction graphs as heuristics to reduce the computational complexity, consider a coalition valid only if the subgraph formed by the agents of the coalition is connected. In this work it as experimentally verify for power law graphs, present in a variety of real graphs, that restricting availability coalition as a connected subgraph may in not prohibited a significant gain. However, they using a subgraphs with strong restrictions, in particular a clique, guarantees an exponential reduction in the number of considered coalition. There no are theorems calculate subgraphs or even the number of cliques on a type power law graph. In the present work it was possible to calculate values experimental for graphs of up to 17 vértices, being also presented analytics results for star graphs (Kn−1,1 ). Star graphs are an acceptable approximation, was they account for hubs, a characteristic structure of power law graphs. As future works can be cited the study of domains where the clique restriction is adequate as well as the development of an algorithm that incorporates the restriction for coalition counting.