Approximating Pareto Sum via Bounded Monotone Min-Plus Convolution

2026-03-26

AI Summary

This work addresses the near-quadratic time bottleneck in computing Pareto-optimal solutions for bicriteria optimization by proposing an efficient additive approximation method. By establishing, for the first time, a fine-grained equivalence between Pareto set approximation and bounded monotone Min-Plus convolution, the problem is reduced to the latter. Building on Chi et al.’s Õ(n^1.5) algorithm, we design a simplified subquadratic approximation scheme that enables flexible trade-offs between runtime and accuracy. Theoretical analysis shows that our method achieves conditionally optimal strongly subquadratic complexity while guaranteeing controllable error bounds. Experimental results demonstrate significant speedups over existing exact algorithms on large-scale instances, substantially reducing runtime while maintaining small output sizes—highlighting the practical utility of our theoretical framework.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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.

Problem

• Pareto sum
• approximation
• bi-criteria optimization
• fine-grained complexity
• Min-Plus Convolution

Innovation

• Pareto sum
• Min-Plus Convolution
• subquadratic algorithm
• fine-grained equivalence
• approximation algorithm