The Multiple Unicast Conjecture is False

2026-10-07

AI Summary

This study addresses whether network coding can surpass the multi-commodity flow throughput limit in undirected multi-unicast networks. By integrating finite-field linear coding, Reed-Solomon local codes, and graph-theoretic constructions, this work proposes a nondeterministic model based on locally verifiable certificates. It further leverages large language models to assist in searching for counterexamples, which are subsequently transformed into deterministic causal codes. The primary contribution lies in successfully refuting the conjecture that network coding offers no advantage over routing, thereby confirming the existence of logarithmic coding gains. These findings establish the significant superiority of network coding in undirected graphs and introduce a novel research paradigm for network information theory.

Abstract

The undirected multiple-unicast conjecture [LL04] asserts that network coding offers no throughput advantage over multicommodity flow. We refute this conjecture by constructing a deterministic linear network code over $\mathbb{F}_9$ on a 182-vertex bipartite subgraph of the point-line incidence graph of $\mathrm{PG}(2,9)$. The construction supports $157$ independent unicast sessions at common coding rate at least $1$, while every fractional multicommodity flow has common rate at most $147/157$. By the amplification theorem of~[BGS17], this yields a family of undirected multiple-unicast instances with coding gap $Ω((\log n)^\varepsilon)$ for some $\varepsilon>0$. We also introduce a nondeterministic model of network coding based on locally verifiable certificates, which guides our construction and may be of independent interest. Building on the high-girth graph and error-correcting code framework of [BH25], we use GPT-6 to find a nondeterministic counterexample based on a new choice of Reed--Solomon local codes, and then convert this example into a causal code using an edge orientation and local search.

Problem

• Multiple Unicast Conjecture
• Network Coding
• Multicommodity Flow
• Coding Gap
• Undirected Networks

Innovation

• Network Coding
• Multiple Unicast Conjecture
• Nondeterministic Model
• Reed-Solomon Codes
• Coding Gap