The zz-matching problem on bipartite graphs

The $z$-matching problem on bipartite graphs is studied with a local algorithm. A $z$-matching ($z \ge 1$) on a bipartite graph is a set of matched edges, in which each vertex of one type is adjacent to at most $1$ matched edge and each vertex of the other type is adjacent to at most $z$ matched edges. The $z$-matching problem on a given bipartite graph concerns finding $z$-matchings with the maximum size. Our approach to this combinatorial optimization are of two folds. From an algorithmic perspective, we adopt a local algorithm as a linear approximate solver to find $z$-matchings on general bipartite graphs, whose basic component is a generalized version of the greedy leaf removal procedure in graph theory. From an analytical perspective, in the case of random bipartite graphs with the same size of two types of vertices, we develop a mean-field theory for the percolation phenomenon underlying the local algorithm, leading to a theoretical estimation of $z$-matching sizes on coreless graphs. We hope that our results can shed light on further study on algorithms and computational complexity of the optimization problem.Comment: 15 pages, 3 figure

Does human imitate successful behaviors immediately?

The emergence and abundance of cooperation in animal and human societies is a challenging puzzle to evolutionary biology. Over the past decades, various mechanisms have been suggested which are capable of supporting cooperation. Imitation dynamics, however, are the most representative microscopic rules of human behaviors on studying these mechanisms. Their standard procedure is to choose the agent to imitate at random from the population. In the spatial version this means a random agent from the neighborhood. Hence, imitation rules do not include the possibility to explore the available strategies, and then they have the possibility to reach a homogeneous state rapidly when the population size is small. To prevent evolution stopping, theorists allow for random mutations in addition to the imitation dynamics. Consequently, if the microscopic rules involve both imitation and mutation, the frequency of agents switching to the more successful strategy must be higher than that of them transiting to the same target strategy via mutation dynamics. Here we show experimentally that the frequency of switching to successful strategy approximates to that of mutating to the same strategy. This suggests that imitation might play an insignificant role on the behaviors of human decision making. In addition, our experiments show that the probabilities of agents mutating to different target strategies are significantly distinct. The actual mutation theories cannot give us an appropriate explanation to the experimental results. Hence, we argue that the mutation dynamics might have evolved for other reasons