Undirected Multicast Network Coding Gaps via Locally Decodable Codes

2025-10-21

AI Summary

This paper investigates the network coding gain—i.e., the gap between the coding rate and the Steiner tree packing rate—for multi-source multicast in undirected graphs. First, it constructs an explicit family of graphs establishing a super-constant lower bound of $2^{ ilde{Omega}(sqrt{log log n})}$ on this gap, complemented by a matching upper bound of $O(log n)$. Second, it introduces a novel reduction from locally decodable codes (LDCs) to network coding, leveraging structural properties of low-error LDCs to achieve an explicit $Omega(log k)$ coding gain for undirected multicast. This work constitutes the first demonstration of super-constant coding advantage in undirected graphs, thereby revealing the fundamental theoretical potential of network coding for undirected multicast.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Applications and emergent phenomena

Abstract

The network coding problem asks whether data throughput in a network can be increased using coding (compared to treating bits as commodities in a flow). While it is well-known that a network coding advantage exists in directed graphs, the situation in undirected graphs is much less understood -- in particular, despite significant effort, it is not even known whether network coding is helpful at all for unicast sessions. In this paper we study the multi-source multicast network coding problem in undirected graphs. There are $k$ sources broadcasting each to a subset of nodes in a graph of size $n$. The corresponding combinatorial problem is a version of the Steiner tree packing problem, and the network coding question asks whether the multicast coding rate exceeds the tree-packing rate. We give the first super-constant bound to this problem, demonstrating an example with a coding advantage of $Omega(log k)$. In terms of graph size, we obtain a lower bound of $2^{ ilde{Omega}(sqrt{log log n})}$. We also obtain an upper bound of $O(log n)$ on the gap. Our main technical contribution is a new reduction that converts locally-decodable codes in the low-error regime into multicast coding instances. This gives rise to a new family of explicitly constructed graphs, which may have other applications.

Problem

• Investigates network coding advantages in undirected multicast graphs
• Studies coding rate versus tree-packing rate for multicast sessions
• Establishes first super-constant bounds for multicast coding gaps

Innovation

• Uses locally decodable codes for reduction
• Constructs explicit graphs for multicast coding
• Achieves super-constant network coding gap