Near-Linear Time Computation of Welzl Orders on Graphs with Linear Neighborhood Complexity

2026-02-16

AI Summary

This work addresses the efficient computation of Welzl orderings with low crossing numbers to accelerate range searching, graph-structure analysis, and first-order model checking. For set systems whose primal and dual shatter functions are linear, the paper presents the first algorithm running in $O(|S| \log |S|)$ time, breaking the previous polynomial-time barrier. The approach integrates random sampling, complexity analysis of set systems, and the linear neighborhood structure of graphs to devise a near-linear-time ordering strategy. On graph classes whose neighborhood complexity is nearly linear, neighborhood covers can be constructed in $O(n \log n)$ time, thereby reducing the time complexity of first-order model checking from $O(n^{5+\varepsilon})$ to $O(n^{3+\varepsilon})$.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Orders with low crossing number, introduced by Welzl, are a fundamental tool in range searching and computational geometry. Recently, they have found important applications in structural graph theory: set systems with linear shatter functions correspond to graph classes with linear neighborhood complexity. For such systems, Welzl's theorem guarantees the existence of orders with only $\mathcal{O}(\log^2 n)$ crossings. A series of works has progressively improved the runtime for computing such orders, from Chazelle and Welzl's original $\mathcal{O}(|U|^3 |\mathcal{F}|)$ bound, through Har-Peled's $\mathcal{O}(|U|^2|\mathcal{F}|)$, to the recent sampling-based methods of Csik\'os and Mustafa. We present a randomized algorithm that computes Welzl orders for set systems with linear primal and dual shatter functions in time $\mathcal{O}(\|S\| \log \|S\|)$, where $\|S\| = |U| + \sum_{X \in \mathcal{F}} |X|$ is the size of the canonical input representation. As an application, we compute compact neighborhood covers in graph classes with (near-)linear neighborhood complexity in time \(\mathcal{O}(n \log n)\) and improve the runtime of first-order model checking on monadically stable graph classes from $\mathcal{O}(n^{5+\varepsilon})$ to $\mathcal{O}(n^{3+\varepsilon})$.

Problem

• Welzl orders
• linear neighborhood complexity
• near-linear time
• set systems
• graph classes

Innovation

• Welzl orders
• near-linear time algorithm
• linear neighborhood complexity
• shatter function
• first-order model checking