Finding diverse solutions in combinatorial problems recently has received considerable attention (Baste et al. 2020; Fomin et al. 2020; Hanaka et al. 2021). In this paper we study the following type of problems: given an integer $k$, the problem asks for $k$ solutions such that the sum of pairwise (weighted) Hamming distances between these solutions is maximized. Such solutions are called diverse solutions. We present a polynomial-time algorithm for finding diverse shortest $st$-paths in weighted directed graphs. Moreover, we study the diverse version of other classical combinatorial problems such as diverse weighted matroid bases, diverse weighted arborescences, and diverse bipartite matchings. We show that these problems can be solved in polynomial time as well. To evaluate the practical performance of our algorithm for finding diverse shortest $st$-paths, we conduct a computational experiment with synthetic and real-world instances.The experiment shows that our algorithm successfully computes diverse solutions within reasonable computational time.

翻译：最近,在分类问题中寻找多种解决办法的问题受到相当重视(Baste等人,2020年;Fomina等人,2020年;Hanaka等人,2021年)。在本文中,我们研究了以下类型的问题:考虑到整数美元,问题要求以k$为单位解决这些解决办法之间的对称(加权)错开距离之和。这些解决办法被称为多种解决办法。我们在加权定向图表中为寻找不同的最起码的美元路径,提出了一个多元时算法。此外,我们研究了其他传统组合问题的不同版本,例如不同的加权甲状腺基、不同的加权芳香和多种双边配对。我们表明这些问题也可以在多元的时间内解决。为了评估我们寻找最短的美元路径的算法的实际表现,我们用合成和现实世界实例进行计算实验。实验表明,我们的算法在合理的计算时间内成功地计算了多种解决办法。