On the k-pairs problem

TL;DR

This paper investigates network coding rates in directed and undirected k-pairs networks, revealing significant gaps between coding rates and meagerness in directed cases and advancing understanding of the k-pairs conjecture in undirected networks.

Contribution

It demonstrates that network coding rates can be substantially smaller than meagerness in directed networks and makes progress on the k-pairs conjecture for undirected networks.

Findings

Network coding rate can be Θ(|V|) smaller than meagerness in directed networks.

Progress made towards the k-pairs conjecture in undirected networks.

Meagerness bounds are tight for directed network coding rates.

Abstract

We consider network coding rates for directed and undirected $k$-pairs networks. For directed networks, meagerness is known to be an upper bound on network coding rates. We show that network coding rate can be $\Theta(|V|)$ multiplicative factor smaller than meagerness. For the undirected case, we show some progress in the direction of the $k$-pairs conjecture.