AI 中文总结
研究由解的大小参数化的多样性问题，引入自然多样性概念，证明若满足一定条件可生成参数化算法，指出多种多样性度量满足自然多样性属性，通过推导相关固定参数算法证明框架适用性，拓宽了参数化多样性算法范围。
AI 中文摘要
多样性优化旨在寻求多个彼此充分不同的高质量解，相较于单个最优解，能更丰富地表示解空间，同时避免完全枚举的高昂成本。在这项工作中，我们引入自然多样性的概念，这是一个连接组合问题$\Pi$和多样性度量$\texttt{dist}$的一般条件。我们表明，如果一对$(\Pi,\texttt{dist})$是自然多样的，且我们有一个完成算法，能从$\Pi$的部分解完成到$\Pi$的一个解，那么当按解的大小和预期解的数量进行参数化时，我们可以生成一个参数化算法来解决关于距离$\texttt{dist}$的$\Pi$的多样性变体。此外，我们表明包括成对不相交性、汉明距离、杰卡德距离和大冢 - 落合系数（在最小和求和聚合下）等几种广泛使用的多样性度量，对于所有解大小相同的问题满足自然多样性属性。最后，我们通过为最小顶点覆盖和最小斯坦纳树的多样变体推导固定参数算法来证明我们框架的适用性。我们的结果通过适应自然解大小参数化和更广泛的多样性度量类别拓宽了参数化多样性算法的范围。
英文摘要
Diversity optimization seeks multiple high-quality solutions that are sufficiently different from one another, providing a richer representation of the solution space than a single optimum while avoiding the prohibitive cost of complete enumeration. In this work, we introduce the notion of natural diversity, a general condition that connects a combinatorial problem $Π$ and a diversity measure $\texttt{dist}$. We show that if a pair $(Π,\texttt{dist})$ is naturally diverse and we have in hand a completion algorithm that, from a partial solution of $Π$, can complete it into a solution of $Π$, then we can produce a parameterized algorithm solving the diversity variant of $Π$ with regard to the distance $\texttt{dist}$ when parameterized by the size of the solutions and the number of expected solutions. Furthermore, we show that several widely used diversity measures, including pairwise disjointness, Hamming distance, Jaccard distance, and the Otsuka-Ochiai coefficient (under both minimum and sum aggregation), satisfy the natural diversity property for problems in which all the solutions have the same size. Finally, we demonstrate the applicability of our framework by deriving fixed-parameter algorithms for diverse variants of Minimum Vertex Cover and Minimum Steiner Tree. Our results broaden the scope of parameterized diversity algorithms by accommodating natural solution-size parameterizations and a wider class of diversity measures.
