Undirected Multicast Network Coding Gaps via Locally Decodable Codes
Mark Braverman, Zhongtian He
TL;DR
This work analyzes the undirected multi-source multicast network coding problem, establishing the first super-constant coding gap: network coding can provide a throughput gain of at least Ω(log k) over noncoding Steiner-based bounds. The authors connect network coding to locally decodable codes, deriving gap constructions from MV-based LDCs and amplifying distance via a binary-tree gadget to achieve a lower bound that translates to n as $2^{\tilde{\Omega}(\sqrt{\log\log n})}$. An accompanying upper bound shows the gap is at most $O(\log n)$, by bounding the throughput with a generalized sparsity measure derived from the dual LP of multi-Steiner packing and leveraging cut-tree packings. The paper also introduces robust-distance LDCs, a relaxed LDC variant enabling practical gap constructions and suggesting a broader applicability of LDC techniques in network coding. Overall, the results illuminate the gap landscape in undirected settings and provide a versatile reduction framework linking coding theory and graph packing, with potential implications for the network coding conjecture and related complexity questions.
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 $Ω(\log k)$. In terms of graph size, we obtain a lower bound of $2^{\tildeΩ(\sqrt{\log \log n})}$. 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.