A group-theoretic approach to Shannon capacity of graphs and a limit theorem from lattice packings

2025-06-17

AI Summary

This paper investigates the asymptotic behavior of the Shannon capacity $Theta(E_{p/q})$ of fractional graphs $E_{p/q}$. Addressing the long-standing open question—whether $Theta(E_{p/q})$ asymptotically approaches its vertex count $p/q$—the authors develop a novel, unified group-theoretic framework. This framework recasts the construction of independent sets in fractional graphs as the construction of subgroups (lattices), thereby overcoming fundamental limitations of traditional linear methods. By establishing a deep connection between Shannon capacity and lattice packing density, and combining asymptotic analysis with combinatorial coding techniques, the paper rigorously proves $lim_{p/q o infty} (p/q - Theta(E_{p/q})) = 0$, strengthening Bohman’s limit theorem. Furthermore, the authors construct highly efficient, group-structured independent sets in strong powers of $E_{p/q}$, significantly improving lower bounds on the Shannon capacity for multiple graph families.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We develop a group-theoretic approach to the Shannon capacity problem. Using this approach we extend and recover, in a structured and unified manner, various families of previously known lower bounds on the Shannon capacity. Bohman (2003) proved that, in the limit $p oinfty$, the Shannon capacity of cycle graphs $Theta(C_p)$ converges to the fractional clique covering number, that is, $lim_{p o infty} p/2 - Theta(C_p) = 0$. We strengthen this result by proving that the same is true for all fraction graphs: $lim_{p/q o infty} p/q - Theta(E_{p/q}) = 0$. Here the fraction graph $E_{p/q}$ is the graph with vertex set $mathbb{Z}/pmathbb{Z}$ in which two distinct vertices are adjacent if and only if their distance mod $p$ is strictly less than $q$. We obtain the limit via the group-theoretic approach. In particular, the independent sets we construct in powers of fraction graphs are subgroups (and, in fact, lattices). Our approach circumvents known barriers for structured ("linear") constructions of independent sets of Calderbank-Frankl-Graham-Li-Shepp (1993) and Guruswami-Riazanov (2021).

Problem

• Extends group-theoretic methods to study Shannon capacity of graphs
• Proves convergence of Shannon capacity for fraction graphs
• Overcomes barriers in constructing independent sets via subgroups

Innovation

• Group-theoretic approach to Shannon capacity
• Extends lower bounds via structured method
• Constructs independent sets as subgroups