Quality diversity~(QD) is a branch of evolutionary computation that gained increasing interest in recent years. The Map-Elites QD approach defines a feature space, i.e., a partition of the search space, and stores the best solution for each cell of this space. We study a simple QD algorithm in the context of pseudo-Boolean optimisation on the ``number of ones'' feature space, where the $i$th cell stores the best solution amongst those with a number of ones in $[(i-1)k, ik-1]$. Here $k$ is a granularity parameter $1 \leq k \leq n+1$. We give a tight bound on the expected time until all cells are covered for arbitrary fitness functions and for all $k$ and analyse the expected optimisation time of QD on \textsc{OneMax} and other problems whose structure aligns favourably with the feature space. On combinatorial problems we show that QD finds a ${(1-1/e)}$-approximation when maximising any monotone sub-modular function with a single uniform cardinality constraint efficiently. Defining the feature space as the number of connected components of a connected graph, we show that QD finds a minimum spanning tree in expected polynomial time.

翻译：质量多样性（QD）是近年来日益受到关注的进化计算分支。Map-Elites QD方法定义一个特征空间（即搜索空间的一个划分），并为该空间中每个单元存储最优解。我们在“1的个数”特征空间上研究伪布尔优化问题中的简单QD算法，其中第i个单元存储属于1的个数在[(i-1)k, ik-1]范围内的最优解，k为粒度参数且满足1 ≤ k ≤ n+1。我们给出了任意适应度函数及所有k值下所有单元被覆盖的期望时间的紧界，并分析了QD算法在OneMax及其他与特征空间结构相契合的问题上的期望优化时间。在组合优化问题中，我们证明当最大化具有单一均匀基数约束的单调子模函数时，QD算法能高效求得{(1-1/e)}-近似解。通过将特征空间定义为连通图的连通分量数，我们证明QD算法可在期望多项式时间内找到最小生成树。