A Framework for the Design of Efficient Diversification Algorithms to NP-Hard Problems

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.