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Evolutionary algorithms and submodular functions: benefits of heavy-tailed mutations
Natural Computing ( IF 2.1 ) Pub Date : 2021-02-16 , DOI: 10.1007/s11047-021-09841-7
Francesco Quinzan , Andreas Göbel , Markus Wagner , Tobias Friedrich

A core operator of evolutionary algorithms (EAs) is the mutation. Recently, much attention has been devoted to the study of mutation operators with dynamic and non-uniform mutation rates. Following up on this area of work, we propose a new mutation operator and analyze its performance on the \((1+1)\) Evolutionary Algorithm (EA). Our analyses show that this mutation operator competes with pre-existing ones, when used by the \((1+1)\) EA on classes of problems for which results on the other mutation operators are available. We show that the \((1+1)\) EA using our mutation operator finds a (1/3)-approximation ratio on any non-negative submodular function in polynomial time. We also consider the problem of maximizing a symmetric submodular function under a single matroid constraint and show that the \((1+1)\) EA using our operator finds a (1/3)-approximation within polynomial time. This performance matches that of combinatorial local search algorithms specifically designed to solve these problems and outperforms them with constant probability. Finally, we evaluate the performance of the \((1+1)\) EA using our operator experimentally by considering two applications: (a) the maximum directed cut problem on real-world graphs of different origins, with up to 6.6 million vertices and 56 million edges and (b) the symmetric mutual information problem using a four month period air pollution data set. In comparison with uniform mutation and a recently proposed dynamic scheme, our operator comes out on top on these instances.



中文翻译:

进化算法和亚模块功能:重尾突变的好处

进化算法(EA)的核心运营商是突变。最近,人们对研究具有动态和非均匀突变率的突变算子进行了大量研究。在此工作领域之后,我们提出了一个新的变异算子,并在\((1 + 1)\)进化算法(EA)上分析了它的性能。我们的分析表明,当\((1 + 1)\)  EA使用该问题算子可以解决其他突变算子的结果时,该突变算子会与现有算子竞争。我们证明\(((1 + 1)\) 使用我们的变异算子的EA在多项式时间内发现任何非负亚模函数的(1/3)逼近率。我们还考虑了在单个拟阵约束下最大化对称子模函数的问题,并表明 使用算子的\((1 + 1)\) EA在多项式时间内找到了(1/3)逼近。这种性能与专门为解决这些问题而设计的组合本地搜索算法相匹配,并且以恒定的概率胜过它们。最后,我们评估\((1 + 1)\)的性能 EA通过考虑以下两种应用,通过实验性地使用我们的运算符:(a)不同起源的真实世界图上的最大有向切割问题,具有多达660万个顶点和5600万个边,以及(b)使用四个月的对称互信息问题期间的空气污染数据集。与统一突变和最近提出的动态方案相比,我们的运算符在这些情况下名列前茅。

更新日期:2021-02-17
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