社会互动作为组合优化问题的启发式算法
Social interaction as a heuristic for combinatorial optimization problems
摘要：
我们调查的阿克塞尔罗德的文化传播模型的一个变体的性能-自适应文化启发式（ACH）-解决NP完全优化问题，即二进制输入模式的大小$F$的布尔二进制感知器的分类。在这种启发式，$N$代理，其特征在于二进制字符串的长度$F$，代表可能的解决方案的优化问题，固定在一个正方形格子的网站和互动与他们最近的邻居。这种相互作用使得代理的字符串（或文化）变得与其邻居的低成本字符串更加相似，从而导致这些字符串在整个晶格中传播。最终的动态冻结成一个均匀的吸收配置，其中所有的代理表现出相同的解决方案的优化问题。我们发现，通过广泛的模拟，找到最佳解决方案的概率是一个功能的减少变量$F/N^{1/4}$，使代理的数量必须增加的问题大小的四次方，$N \propto F^ 4$，以保证一个固定的成功概率。在这种情况下，我们发现达到吸收配置的弛豫时间与$F^ 6$成比例，这可以被解释为ACH的总计算成本，以找到一组最佳的权重为布尔二进制感知器，给定一个固定的成功概率。
英文摘要：
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 $F$ by a Boolean Binary Perceptron. In this heuristic, $N$ agents, characterized by binary strings of length $F$ 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 $F/N^{1/4}$ so that the number of agents must increase with the fourth power of the problem size, $N \propto F^ 4$, to guarantee a fixed probability of success. In this case, we find that the relaxation time to reach an absorbing configuration scales with $F^ 6$ 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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