arXiv 1612.069622016-12cs.DScs.AIcs.NE

TSP问题交叉熵算法的随机性分析

Stochastic Runtime Analysis of a Cross Entropy Algorithm for Traveling Salesman Problems

Beijing Institute for Scientific and Engineering Computing（北京科学与工程计算研究院）；Beijing University of Technology（北京工业大学）；Berlin University of Technology（柏林工业大学）

Zijun Wu, Rolf Moehring, Jianhui Lai

备注：: 40 pages, 7 figures
发表信息：: Theoretical Computer Science, 2017
DOI：: 10.1016/j.tcs.2017.10.012

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摘要：

分析了两类旅行商问题的交叉熵算法的随机运行时间。该算法的主要特点是著名的最大最小蚂蚁系统的迭代最佳增强。对于简单的情况下，有一个$\{1,n\}$值的距离函数和一个唯一的最优解，我们证明了随机运行时间的$O(n^{6+ε})$与基于顶点的随机解生成，和随机运行时间的$O(n^{3+ε}\ln n)$与基于边的随机解生成任意$ε\in (0,1)$。这些运行时间非常接近已知的最大最小蚂蚁系统的变体的预期运行时间，具有迄今为止最好的强化。它们是针对随机运行时间的更强概念获得的，这意味着在该时间内以压倒性的概率获得最优解，即，随着问题规模的增长，概率以指数方式快速趋于1。我们还检查了$n$顶点位于$m\times m$网格上的更复杂的实例。当$n$顶点跨越凸多边形时，我们得到了基于顶点的随机解生成的$O(n^{3}m^{5+ε})$的随机运行时间，以及基于边的随机解生成的$O(n^{2}m^{5+ε})$的随机运行时间.当一个凸多边形内有多个$k = O(1)$顶点时，通过基于顶点的随机解生成，我们得到了$O(n^{4}m^{5+ε}+n^{6k-1}m^ε)$的随机运行时间;通过基于边的随机解生成，我们得到了$O(n^{3}m^{5+ε}+n^{3k}m^ε)$的随机运行时间.这些运行时比在最近的一篇文章中报道的所谓的$(μ\!+\!λ)$ EA的预期运行时更好，并且再次获得了更强的随机运行时概念。

英文摘要：

This article analyzes the stochastic runtime of a Cross-Entropy Algorithm on two classes of traveling salesman problems. The algorithm shares main features of the famous Max-Min Ant System with iteration-best reinforcement. For simple instances that have a $\{1,n\}$-valued distance function and a unique optimal solution, we prove a stochastic runtime of $O(n^{6+ε})$ with the vertex-based random solution generation, and a stochastic runtime of $O(n^{3+ε}\ln n)$ with the edge-based random solution generation for an arbitrary $ε\in (0,1)$. These runtimes are very close to the known expected runtime for variants of Max-Min Ant System with best-so-far reinforcement. They are obtained for the stronger notion of stochastic runtime, which means that an optimal solution is obtained in that time with an overwhelming probability, i.e., a probability tending exponentially fast to one with growing problem size. We also inspect more complex instances with $n$ vertices positioned on an $m\times m$ grid. When the $n$ vertices span a convex polygon, we obtain a stochastic runtime of $O(n^{3}m^{5+ε})$ with the vertex-based random solution generation, and a stochastic runtime of $O(n^{2}m^{5+ε})$ for the edge-based random solution generation. When there are $k = O(1)$ many vertices inside a convex polygon spanned by the other $n-k$ vertices, we obtain a stochastic runtime of $O(n^{4}m^{5+ε}+n^{6k-1}m^ε)$ with the vertex-based random solution generation, and a stochastic runtime of $O(n^{3}m^{5+ε}+n^{3k}m^ε)$ with the edge-based random solution generation. These runtimes are better than the expected runtime for the so-called $(μ\!+\!λ)$ EA reported in a recent article, and again obtained for the stronger notion of stochastic runtime.

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