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Social interaction as a heuristic for combinatorial optimization problems
We investigate the performance of a variant of Axelrod's model for
dissemination of culture - the Adaptive Culture Heuristic (ACH) - on solving an
NP-Complete optimization problem, namely, the classification of binary input
patterns of size by a Boolean Binary Perceptron. In this heuristic,
agents, characterized by binary strings of length which represent possible
solutions to the optimization problem, are fixed at the sites of a square
lattice and interact with their nearest neighbors only. The interactions are
such that the agents' strings (or cultures) become more similar to the low-cost
strings of their neighbors resulting in the dissemination of these strings
across the lattice. Eventually the dynamics freezes into a homogeneous
absorbing configuration in which all agents exhibit identical solutions to the
optimization problem. We find through extensive simulations that the
probability of finding the optimal solution is a function of the reduced
variable so that the number of agents must increase with the fourth
power of the problem size, , to guarantee a fixed probability
of success. In this case, we find that the relaxation time to reach an
absorbing configuration scales with which can be interpreted as the
overall computational cost of the ACH to find an optimal set of weights for a
Boolean Binary Perceptron, given a fixed probability of success
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