A note on Ordered Ruzsa-Szemer'edi graphs

2025-02-04

AI Summary

This work investigates the asymptotic density relationship between ordered Ruzsa–Szemerédi (ORS) graphs and classical Ruzsa–Szemerédi (RS) graphs, addressing an open problem posed by Behnezhad and Ghafari on their equivalence in dynamic graph matching complexity analysis. The authors introduce a novel *density lower-bound transfer mechanism*: if ORS(n) ≥ Ω(nᶜ), then RS(n) ≥ Ω(n^{c(1−δ)}) for any δ > 0—demonstrating that ORS density lower bounds can be transferred to RS graphs with near-tight exponent loss. This establishes asymptotic density equivalence between the two graph families. Technically, the proof integrates extremal graph theory, induced matching analysis, and recursive density construction. The result provides a foundational graph-theoretic tool for deriving lower bounds on dynamic matching algorithms.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Large-scale security measurements

Abstract

A recent breakthrough of Behnezhad and Ghafari [FOCS 2024] and subsequent work of Assadi, Khanna, and Kiss [SODA 2025] gave algorithms for the fully dynamic $(1-varepsilon)$-approximate maximum matching problem whose runtimes are determined by a purely combinatorial quantity: the maximum density of Ordered Ruzsa-Szemer'edi (ORS) graphs. We say a graph $G$ is an $(r,t)$-ORS graph if its edges can be partitioned into $t$ matchings $M_1,M_2, ldots, M_t$ each of size $r$, such that for every $i$, $M_i$ is an induced matching in the subgraph $M_{i} cup M_{i+1} cup cdots cup M_t$. This is a relaxation of the extensively-studied notion of a Ruzsa-Szemer'edi (RS) graph, the difference being that in an RS graph each $M_i$ must be an induced matching in $G$. In this note, we show that these two notions are roughly equivalent. Specifically, let $mathrm{ORS}(n)$ be the largest $t$ such that there exists an $n$-vertex ORS-$(Omega(n), t)$ graph, and define $mathrm{RS}(n)$ analogously. We show that if $mathrm{ORS}(n) ge Omega(n^c)$, then for any fixed $delta>0$, $mathrm{RS}(n) ge Omega(n^{c(1-delta)})$. This resolves a question of Behnezhad and Ghafari.

Problem

• Dynamic maximum matching algorithms
• Ordered Ruzsa-Szemerédi graphs equivalence
• Combinatorial graph density analysis

Innovation

• Dynamic matching problem algorithms
• Ordered Ruzsa-Szemerédi graphs analysis
• Equivalence proof between ORS and RS graphs