The Pareto sum of two-dimensional point sets $P$ and $Q$ in $\mathbb{R}^2$ is defined as the skyline of the points in their Minkowski sum. The problem of efficiently computing the Pareto sum arises frequently in bi-criteria optimization algorithms. Prior work establishes that computing the Pareto sum of sets $P$ and $Q$ of size $n$ suffers from conditional lower bounds that rule out strongly subquadratic $O(n^{2-ε})$-time algorithms, even when the output size is $Θ(n)$. Naturally, we ask: How efficiently can we \emph{approximate} Pareto sums, both in theory and practice? Can we beat the near-quadratic-time state of the art for exact algorithms? On the theoretical side, we formulate a notion of additively approximate Pareto sets and show that computing an approximate Pareto set is \emph{fine-grained equivalent} to Bounded Monotone Min-Plus Convolution. Leveraging a remarkable $\tilde{O}(n^{1.5})$-time algorithm for the latter problem (Chi, Duan, Xie, Zhang; STOC '22), we thus obtain a strongly subquadratic (and conditionally optimal) approximation algorithm for computing Pareto sums. On the practical side, we engineer different algorithmic approaches for approximating Pareto sets on realistic instances. Our implementations enable a granular trade-off between approximation quality and running time/output size compared to the state of the art for exact algorithms established in (Funke, Hespe, Sanders, Storandt, Truschel; Algorithmica '25). Perhaps surprisingly, the (theoretical) connection to Bounded Monotone Min-Plus Convolution remains beneficial even for our implementations: in particular, we implement a simplified, yet still subquadratic version of an algorithm due to Chi, Duan, Xie and Zhang, which on some sufficiently large instances outperforms the competing quadratic-time approaches.

翻译：二维点集 $P$ 和 $Q$（位于 $\mathbb{R}^2$ 中）的帕累托和定义为它们闵可夫斯基和中的天际线点。在双目标优化算法中，高效计算帕累托和的需求频繁出现。先前的工作表明，当输出规模为 $Θ(n)$ 时，大小为 $n$ 的点集 $P$ 和 $Q$ 的帕累托和计算受到条件性下界的限制，排除了强次二次 $O(n^{2-ε})$-时间算法的存在。自然，我们问：在理论和实践中，我们能以多高的效率\emph{近似}帕累托和？能否超越精确算法的近二次时间最新技术水平？在理论方面，我们提出了加法近似帕累托集的概念，并证明计算近似帕累托集与有界单调最小加卷积问题\emph{精细等价}。利用针对后一问题的显著 $\tilde{O}(n^{1.5})$-时间算法（Chi, Duan, Xie, Zhang; STOC '22），我们由此获得了一个强次二次（且条件最优）的帕累托和计算近似算法。在实践方面，我们设计了多种算法方法来近似实际实例中的帕累托集。与 (Funke, Hespe, Sanders, Storandt, Truschel; Algorithmica '25) 中建立的精确算法最新技术水平相比，我们的实现使得近似质量、运行时间/输出规模之间能够实现精细权衡。也许令人惊讶的是，（理论上的）与有界单调最小加卷积的联系对我们的实现仍然有益：特别是，我们实现了 Chi、Duan、Xie 和 Zhang 算法的一个简化但仍是次二次的版本，该版本在足够大的实例上优于竞争性的二次时间方法。