帕累托集的帕累托和：下界与算法
Pareto Sums of Pareto Sets: Lower Bounds and Algorithms
AI总结：
针对双准则优化中帕累托集合并需求，本文提出多种高效帕累托和计算算法，证明条件下界，优化工程实现，并通过实验及道路网路径规划应用验证有效性。
摘要：
在双准则优化问题中，目标通常是计算帕累托最优解集。这类问题的许多算法都依赖于对部分解的高效合并或组合，以及对结果集中被支配解的过滤。本文研究了计算两个大小均为n的给定帕累托集A、B的帕累托和的任务。帕累托和C包含闵可夫斯基和M = {a+b|a ∈ A, b∈B}中的所有非支配点。由于闵可夫斯基和的大小为n²，但帕累托和C的规模可能小得多，因此目标是无需计算并存储全部M即可算出C。\n我们提出了若干用于高效计算帕累托和的新算法，包括一种输出敏感的逐次算法，其运行时间为O(n log n + nk)，空间复杂度为O(n+k)，其中k=|C|。若C的元素以流的形式输入，空间复杂度可降至O(n)。对于输出规模k ≥ 2n的情况，我们证明了帕累托和计算的一个条件下界：除非(min,+)-卷积硬度猜想不成立，否则不存在运行时间为O(n^(2-δ))（δ>0）的算法。当k ∈ Θ(n)时，该逐次算法的运行时间与该下界匹配。然而，当k ∈ Θ(n²)时，逐次算法的运行时间为三次方级别。但我们也提出了一种具有输出敏感空间复杂度的算法，其运行时间为O(n² log n)，即使对于大k值，其运行时间也仅相差一个对数因子即可匹配下界。\n此外，我们还介绍了合适的工程优化技术，以提升我们算法的实际运行效率。最后，我们在生成数据和真实世界数据上开展了广泛的对比实验研究。作为展示性应用，我们研究了道路网络中基于预处理的双准则路径规划问题。
英文摘要：
In bi-criteria optimization problems, the goal is typically to compute the set of Pareto-optimal solutions. Many algorithms for these types of problems rely on efficient merging or combining of partial solutions and filtering of dominated solutions in the resulting sets. In this article, we consider the task of computing the Pareto sum of two given Pareto sets $A, B$ of size $n$. The Pareto sum $C$ contains all non-dominated points of the Minkowski sum $M = \{a+b|a \in A, b\in B\}$. Since the Minkowski sum has a size of $n^2$, but the Pareto sum $C$ can be much smaller, the goal is to compute $C$ without having to compute and store all of $M$. We present several new algorithms for efficient Pareto sum computation, including an output-sensitive successive algorithm with a running time of $O(n \log n + nk)$ and a space consumption of $O(n+k)$ for $k=|C|$. If the elements of $C$ are streamed, the space consumption reduces to $O(n)$. For output sizes $k \geq 2n$, we prove a conditional lower bound for Pareto sum computation, which excludes running times in $O(n^{2-δ})$ for $δ> 0$ unless the (min,+)-convolution hardness conjecture fails. The successive algorithm matches this lower bound for $k \in Θ(n)$. However, for $k \in Θ(n^2)$, the successive algorithm exhibits a cubic running time. But we also present an algorithm with an output-sensitive space consumption and a running time of $O(n^2 \log n)$, which matches the lower bound up to a logarithmic factor even for large $k$. Furthermore, we describe suitable engineering techniques to improve the practical running times of our algorithms. Finally, we provide an extensive comparative experimental study on generated and real-world data. As a showcase application, we consider preprocessing-based bi-criteria route planning in road networks.
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