Approximating Pareto Sum via Bounded Monotone Min-Plus Convolution

Abstract

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.

Figures (16)

• Figure 1: Two Pareto sets $P,Q$ on the left arising from bi-criteria route planning. Depicted on the right is the Minkowski sum $P+Q$ with the Pareto sum $PS(P,Q)$.
• Figure 2: Reduction from Pareto sum to bounded monotone min-plus convolution and back-transformation of the solution on a small example.
• Figure 3: The points $\tilde{s}_1$ and $\tilde{s}_2$ are examples of points with $d(\tilde{s}, s) \leq \Delta$ for the shown $s \in S$. Note that if $s\in S$, then no point in $P+Q$ dominates $s$ ("i.e.", no point lies to the lower left of $s$).
• Figure 4: Overview of the main steps of our additive approximation algorithms. An important step is to compute the Pareto sum of the scaled instance. This can either be done directly via a Pareto sum algorithm or by an algorithm for bounded monotone min-plus convolution when applying the proper forward and backward reduction steps.
• Figure 5: Two different versions of generating input sets with values in the range $[0,2n]$: Near-linear and near-curved input sets are based on the linear and curve generator, respectively, where we apply perturbation to a subset of the points.

Theorems & Definitions (25)

• Definition 1: Bounded Pareto Sum
• Definition 2: Bounded Monotone Min-Plus Convolution
• Theorem 3
• Corollary 4
• Definition 5: $\Delta$-Approximative Pareto sum
• Lemma 6
• Theorem 7
• Corollary 8
• Theorem 9
• Remark 10