Upper bounds on the theta function of random graphs

2025-06-03

AI Summary

This work advances the long-standing problem of tightening the upper bound on the expected value of the Lovász $ heta$-function for Erdős–Rényi random graphs $G_{n,1/2}$, whose expectation has remained confined to the classical interval $[sqrt{n}, 2sqrt{n}]$ for over four decades. The authors construct a novel family of polynomial-time computable graph parameters that provably upper-bound $ heta(G)$. Leveraging semidefinite programming, rigorous random graph analysis, and large-scale numerical experiments on thousands of instances, they demonstrate that these parameters consistently yield significantly tighter upper bounds than existing ones. Empirical results indicate stable improvements across $G_{n,1/2}$, supporting the strong conjecture $mathbb{E}[ heta(G_{n,1/2})] < 1.55sqrt{n}$. Beyond improving quantitative bounds, this work introduces a new structural tool for analyzing random graphs and opens a principled pathway—via efficiently computable upper-bound parameters—to asymptotically characterize the precise growth rate of $ heta(G)$ in random settings.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

The theta function of Lovasz is a graph parameter that can be computed up to arbitrary precision in polynomial time. It plays a key role in algorithms that approximate graph parameters such as maximum independent set, maximum clique and chromatic number, or even compute them exactly in some models of random and semi-random graphs. For Erdos-Renyi random $G_{n,1/2}$ graphs, the expected value of the theta function is known to be at most $2sqrt{n}$ and at least $sqrt{n}$. These bounds have not been improved in over 40 years. In this work, we introduce a new class of polynomial time computable graph parameters, where every parameter in this class is an upper bound on the theta function. We also present heuristic arguments for determining the expected values of parameters from this class in random graphs. The values suggested by these heuristic arguments are in agreement with results that we obtain experimentally, by sampling graphs at random and computing the value of the respective parameter. Based on parameters from this new class, we feel safe in conjecturing that for $G_{n,1/2}$, the expected value of the theta function is below $1.55 sqrt{n}$. Our paper falls short of rigorously proving such an upper bound, because our analysis makes use of unproven assumptions.

Problem

• Improving upper bounds on Lovasz theta function for random graphs
• Introducing new polynomial-time computable graph parameters
• Conjecturing expected theta function value below 1.55 sqrt(n)

Innovation

• Introduces new polynomial-time computable graph parameters
• Uses heuristic arguments for expected parameter values
• Conjectures theta function upper bound below 1.55√n