There has been considerable recent interest in computing a diverse collection of solutions to a given optimization problem. Given a classical optimization problem $Π$ (e.g., minimum spanning tree, maximum independent set) with input size $n$ and an integer $k\geq 1$, the goal is to generate a collection of $k$ maximally diverse solutions to $Π$. 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 achieving energy efficiency. For several problems $Π$ solvable in polynomial time, there are efficient $\text{poly}(n,k)$-time exact or approximation algorithms available for their diverse-solution variants. In contrast, only FPT algorithms are known for NP-complete problems such as vertex covers and independent sets. In the worst case, these algorithms run in time $\exp((kn)^a)$ for some $a>0$. In this work, we address this gap and give a general framework for $\text{poly}(n,k)$-time \emph{bi-criteria} PTAS for the diverse variants of several NP-hard problems such as \textsc{Knapsack}, \textsc{Maximum Weight Independent Set} (MWIS) and \textsc{Minimum Weight Vertex Cover} (MWVC) in planar graphs. More precisely, given a maximization (or, minimization) problem $Π$, an integer $k \geq 1$, user-specified quality threshold $ c \in (0, 1) $, and user-controlled error parameters $δ,\varepsilon \in (0,1)$, our bi-approximation algorithm returns a collection of $k$ solutions $S_1,S_2,\ldots,S_k$ that are $(1-δ)c$-optimal and whose diversity is at least a $(1-\varepsilon)$-fraction of the maximum diversity achievable by any collection of $k$ $c$-optimal solutions. To complement our positive results, we prove that the diverse-solution variants of \KP and MWIS mentioned above do not admit standard single-criterion PTASs.

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