An Optimization Approach to the Langberg-M\'edard Multiple Unicast Conjecture

TL;DR

This paper formulates an optimization approach to the Langberg-Médard conjecture, demonstrating asymptotic optimality of previous solutions and proposing a perturbation framework that improves multi-flow rates for certain network sizes.

Contribution

It introduces an optimization framework that refines previous multi-flow constructions and provides improved solutions for specific network sizes, advancing understanding of the conjecture.

Findings

Previous construction yields asymptotically optimal solutions.

Perturbation framework improves multi-flow rates for certain k values.

Achieves the largest multi-flow rates to date for k=3 to 10.

Abstract

The Langberg-M\'edard multiple unicast conjecture claims that for any strongly reachable $k$-pair network, there exists a multi-flow with rate $(1,1,\dots,1)$. In a previous work, through combining and concatenating the so-called elementary flows, we have constructed a multi-flow with rate at least $(\frac{8}{9}, \frac{8}{9}, \dots, \frac{8}{9})$ for any $k$. In this paper, we examine an optimization problem arising from this construction framework. We first show that our previous construction yields a sequence of asymptotically optimal solutions to the aforementioned optimization problem. And furthermore, based on this solution sequence, we propose a perturbation framework, which not only promises a better solution for any $k \mod 4 \neq 2$ but also solves the optimization problem for the cases $k=3, 4, \dots, 10$, accordingly yielding multi-flows with the largest rate to date.