There has been considerable recent interest in computing a diverse collection of solutions to a given optimization problem, both in the AI and theory communities. Given a classical optimization problem $\Pi$ (e.g., spanning tree, minimum cuts, maximum matching, minimum vertex cover) with input size $n$ and an integer $k\geq 1$, the goal is to generate a collection of $k$ maximally diverse solutions to $\Pi$. This diverse-X paradigm not only allows the user to generate very different solutions, but also helps make systems more secure and robust by handling uncertainty, and achieve energy efficiency. For problems $\Pi$ in P (such as spanning tree and minimum cut), there are efficient $\text{poly}(n,k)$ approximation algorithms available for the diverse variants [Hanaka et al. AAAI 2021, 2022, 2023, Gao et al. LATIN 2022, de Berg et al. ISAAC 2023]. In contrast, only FPT algorithms are known for NP-hard problems such as vertex covers and independent sets [Baste et al. IJCAI 2020, Eiben et al. SODA 2024, Misra et al. ISAAC 2024, Austrin et al. ICALP 2025], but in the worst case, these algorithms run in time $\exp((kn)^c)$ for some $c>0$. In this work, we address this gap and give $\text{poly}(n,k)$ or $f(k)\text{poly}(n)$ time approximation algorithms for diversification variants of several NP-hard problems such as knapsack, maximum weight independent sets (MWIS) and minimum vertex covers in planar graphs, geometric (rectangle) knapsack, enclosing points by polygon, and MWIS in unit-disk-graphs of points in convex position. Our results are achieved by developing a general framework and applying it to problems with textbook dynamic-programming algorithms to find one solution.

翻译：近年来，在人工智能和理论计算机科学领域，计算给定优化问题多样化解集的研究引起了广泛关注。给定一个经典优化问题Π（例如生成树、最小割、最大匹配、最小顶点覆盖），其输入规模为n且整数k≥1，目标是生成Π的k个最大程度多样化的解集。这种多样化-X范式不仅使用户能够生成差异显著的解，还有助于通过处理不确定性使系统更具安全性和鲁棒性，并实现能效优化。对于P类问题（如生成树和最小割），其多样化变体存在高效的poly(n,k)近似算法[Hanaka等人AAAI 2021, 2022, 2023, Gao等人LATIN 2022, de Berg等人ISAAC 2023]。相比之下，对于NP难问题（如顶点覆盖和独立集），目前仅知FPT算法[Baste等人IJCAI 2020, Eiben等人SODA 2024, Misra等人ISAAC 2024, Austrin等人ICALP 2025]，但在最坏情况下这些算法的运行时间为exp((kn)^c)（其中c>0）。本研究致力于填补这一空白，针对多个NP难问题的多样化变体（如背包问题、平面图中最大权独立集和最小顶点覆盖、几何（矩形）背包问题、多边形包围点问题，以及凸位置点集单位圆图的最大权独立集），提出了poly(n,k)或f(k)poly(n)时间近似算法。我们的成果通过开发通用框架实现，并将其应用于具有教科书式动态规划算法（用于寻找单一解）的问题。