On the Multiple-Unicast Conjecture: Beyond Cut Metrics

2026-08-06

AI Summary

This work investigates whether network coding outperforms fractional routing in multiple-unicast sessions over undirected networks—the so-called “multiple-unicast conjecture.” To this end, the authors propose a unified analytical framework based on graph metrics, generalizing the traditional cut-set bound to broader graph structures as the foundation for performance comparison. Within this framework, they establish—without computer assistance—the validity of the conjecture for three new classes of networks: those with at most five terminals, certain planar networks, and arbitrarily large networks satisfying specific endpoint structural constraints. Furthermore, they demonstrate a logical connection between the Γ₃,₃ instance and the general case, offering new insights into the conjecture’s broader applicability.

Abstract

Network coding allows intermediate nodes to encode received messages before transmission. The multiple-unicast conjecture asserts that coding has no throughput advantage over fractional routing for independent unicast sessions in any undirected network. Despite more than two decades of sustained study, this central open problem remains unresolved. The conjecture is deeply connected to computational complexity: a proof would yield long-sought lower bounds for fundamental problems. To study the conjecture, this paper develops a unified metric framework from the perspective that the basic objects behind the comparison between coding and routing are not cuts alone, but graph metrics. Using this framework, we prove the conjecture for three new classes of undirected networks: (a) networks with at most five terminal locations; (b) planar networks whose terminal locations lie on the boundaries of at most three designated faces, with each session's endpoints on one such face; and (c) networks with arbitrarily many nodes and terminal locations under a structural restriction on session endpoints. We give a new proof that the conjecture holds for networks with at most six coding nodes, without computer-aided search, and show that if the conjecture holds on $Γ_{3,3}$, then it holds whenever no three sessions have six distinct terminal locations.

Problem

• multiple-unicast conjecture
• network coding
• fractional routing
• undirected networks
• throughput advantage

Innovation

• network coding
• multiple-unicast conjecture
• graph metrics
• fractional routing
• undirected networks