# Network Coding Gaps for Completion Times of Multiple Unicasts

## TL;DR

This paper investigates the potential benefits of network coding over routing for minimizing completion times in multiple unicast networks, establishing bounds on the coding gap that are polylogarithmic in the number of unicast sessions.

## Contribution

It introduces new techniques to upper bound the network coding gap for makespan minimization in multiple unicasts, showing it is at most polylogarithmic in the number of sessions.

## Key findings

- Network coding gap is at most polylogarithmic in the number of unicasts.
- Existence of instances where the coding gap is polylogarithmic in the number of unicasts.
- Results extend to average completion time and any  norm of completion times.

## Abstract

We study network coding gaps for the problem of makespan minimization of multiple unicasts. In this problem distinct packets at different nodes in a network need to be delivered to a destination specific to each packet, as fast as possible. The network coding gap specifies how much coding packets together in a network can help compared to the more natural approach of routing. While makespan minimization using routing has been intensely studied for the multiple unicasts problem, no bounds on network coding gaps for this problem are known. We develop new techniques which allow us to upper bound the network coding gap for the makespan of $k$ unicasts, proving this gap is at most polylogarithmic in $k$. Complementing this result, we show there exist instances of $k$ unicasts for which this coding gap is polylogarithmic in $k$. Our results also hold for average completion time, and more generally any $\ell_p$ norm of completion times.