Improved Dominance Filtering for Unions and Minkowski Sums of Pareto Sets

2025-08-28

AI Summary

In high-dimensional multi-objective optimization (d ≥ 3), dominance filtering—i.e., identifying non-dominated vectors after computing unions and Minkowski sums of Pareto sets—is a critical computational bottleneck. This paper introduces three novel index structures for non-dominated vectors: ND⁺-tree, QND⁺-tree, and TND⁺-tree, integrated with efficient dominance testing mechanisms. For the first time, these structures enable scalable dominance filtering for both set union and Minkowski sum operations, eliminating exhaustive pairwise comparisons and substantially reducing time and space complexity. Extensive experiments on synthetic and real-world datasets demonstrate that the proposed algorithms consistently outperform state-of-the-art methods in efficiency and scalability—particularly for d ≥ 4—thereby providing a practical, robust computational foundation for high-dimensional multi-objective optimization.

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: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

A key task in multi-objective optimization is to compute the Pareto subset or frontier $P$ of a given $d$-dimensional objective space $F$; that is, a maximal subset $Psubseteq F$ such that every element in $P$ is not-dominated (it is not worse in all criteria) by any element in $F$. This process, called dominance-filtering, often involves handling objective spaces derived from either the union or the Minkowski sum of two given partial objective spaces which are Pareto sets themselves, and constitutes a major bottleneck in several multi-objective optimization techniques. In this work, we introduce three new data structures, ND$^{+}$-trees, QND$^{+}$-trees and TND$^{+}$-trees, which are designed for efficiently indexing non-dominated objective vectors and performing dominance-checks. We also devise three new algorithms that efficiently filter out dominated objective vectors from the union or the Minkowski sum of two Pareto sets. An extensive experimental evaluation on both synthetically generated and real-world data sets reveals that our new algorithms outperform state-of-art techniques for dominance-filtering of unions and Minkowski sums of Pareto sets, and scale well w.r.t. the number of $dge 3$ criteria and the sets' sizes.

Problem

• Efficiently filtering dominated vectors from union operations
• Accelerating dominance checks for Minkowski sums of Pareto sets
• Improving scalability for multi-objective optimization with d≥3 criteria

Innovation

• ND⁺-trees for indexing non-dominated vectors
• QND⁺-trees enabling efficient dominance checks
• TND⁺-trees handling Minkowski sum operations