A Session Interaction Framework for The Multiple-Unicast Conjecture

2026-08-06

AI Summary

This study investigates the validity of the multiple-unicast conjecture in undirected networks—namely, whether network coding cannot surpass the throughput limits achievable by routing. To this end, the authors propose a session interaction framework that decomposes verification into two stages: first simplifying the session set via session dominance relations, then applying a session decoupling theorem to the resulting irreducible core to partition it into independent subsets for individual validation. The key contribution lies in establishing, for the first time, an equivalence between the multiple-unicast conjecture and the topological independence of the irreducible core. Furthermore, the work introduces geometric simplification conditions and a high-cost cut-set criterion, yielding a computable and iterative verification mechanism applicable to arbitrary undirected networks, thereby advancing the intersection of network information theory and computational complexity theory.

Abstract

The multiple-unicast conjecture asserts that network coding offers no throughput advantage over routing in undirected networks. Its validity is known to imply fundamental lower bounds in computational complexity. We propose a Session Interaction Framework that reduces the conjecture to a central equivalence: the conjecture holds universally if and only if every irreducible core is independent. This result transforms the global feasibility problem into a two-stage process. First, to make the reduction phase tractable, we provide simplified sufficient conditions for session dominance, offering geometric criteria to iteratively simplify complex session sets. Second, for the remaining "irreducible core," we propose a Session Decoupling Theorem, reducing the conjecture's validity for a session set to its independent subsets. Topologically, we prove that sessions separated by high-cost cuts or cut-vertices are guaranteed to be independent. By integrating these reduction and decomposition mechanisms, our framework offers a systematic methodology to verify the conjecture across general network topologies.

Problem

• multiple-unicast conjecture
• network coding
• undirected networks
• throughput advantage
• session independence

Innovation

• Session Interaction Framework
• Multiple-Unicast Conjecture
• Irreducible Core
• Session Decoupling Theorem
• Network Coding