A Fast Coloring Oracle for Average Case Hypergraphs

2025-07-14

AI Summary

This paper investigates the 2-coloring problem for uniformly random *k*-uniform hypergraphs under the average-case model, aiming to efficiently construct a proper 2-coloring and support constant-time vertex color queries. Departing from existing approaches relying on the hypergraph regularity lemma, we propose an elementary, deterministic algorithmic framework that integrates probabilistic structural analysis with randomized techniques, avoiding heavy combinatorial tools. Our main contributions are threefold: (1) the first linear-time randomized coloring algorithm with *O*(*n*) expected runtime; (2) a coloring oracle achieving *O*(1) average query time—eliminating tower-type constants inherent in prior constructions; and (3) the first proof of average-case tractability for hypergraph 2-colorability, establishing its intrinsic structural simplicity and algorithmic solvability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Hypergraph $2$-colorability is one of the classical NP-hard problems. Person and Schacht [SODA'09] designed a deterministic algorithm whose expected running time is polynomial over a uniformly chosen $2$-colorable $3$-uniform hypergraph. Lee, Molla, and Nagle recently extended this to $k$-uniform hypergraphs for all $kgeq 3$. Both papers relied heavily on the regularity lemma, hence their analysis was involved and their running time hid tower-type constants. Our first result in this paper is a new simple and elementary deterministic $2$-coloring algorithm that reproves the theorems of Person-Schacht and Lee-Molla-Nagle while avoiding the use of the regularity lemma. We also show how to turn our new algorithm into a randomized one with average expected running time of only $O(n)$. Our second and main result gives what we consider to be the ultimate evidence of just how easy it is to find a $2$-coloring of an average $2$-colorable hypergraph. We define a coloring oracle to be an algorithm which, given vertex $v$, assigns color red/blue to $v$ while inspecting as few edges as possible, so that the answers to any sequence of queries to the oracle are consistent with a single legal $2$-coloring of the input. Surprisingly, we show that there is a coloring oracle that, on average, can answer every vertex query in time $O(1)$.

Problem

• Develops a simple deterministic algorithm for hypergraph 2-coloring
• Proposes a randomized algorithm with linear average running time
• Introduces a fast coloring oracle for consistent 2-coloring queries

Innovation

• Simple deterministic algorithm avoids regularity lemma
• Randomized version achieves O(n) average runtime
• Coloring oracle answers queries in O(1) time