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（Quality Diversity，QD）是进化计算的一个分支，近年来日益受到关注。Map-Elites QD方法定义了一个特征空间（即对搜索空间进行划分），并为该空间的每个单元存储最优解。我们在“1的个数”特征空间中研究简单的QD算法在伪布尔优化问题上的表现，其中第$i$个单元存储所有包含$[(i-1)k, ik-1]$个1的解中的最优解。这里$k$是粒度参数，满足$1 \leq k \leq n+1$。我们给出了在任意适应度函数下以及所有$k$值中覆盖所有单元的期望时间的紧确界，并分析了QD在\textsc{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QD在最大化具有单一均匀基数约束的任意单调子模函数时，能高效地找到${(1-1/e)}$-近似解。当特征空间定义为连通图的连通分量数量时，我们证明QD能在期望多项式时间内找到最小生成树。