# Centralized Network Utility Maximization over Aggregate Flows

## TL;DR

This paper presents a decomposition approach for network utility maximization that simplifies complex flow rate problems into aggregate flow problems, with solutions applicable to both single-path and multipath scenarios, using ADMM.

## Contribution

It introduces a novel decomposition theorem for NUM problems based on source-destination aggregation, applicable to various utility functions and implementable via ADMM.

## Key findings

- Theorems relating detailed flow rates to aggregate flows for NUM.
- A simple apportionment method for $\
- Implementation of the decomposition using ADMM.

## Abstract

We study a network utility maximization (NUM) decomposition in which the set of flow rates is grouped by source-destination pairs. We develop theorems for both single-path and multipath cases, which relate an arbitrary NUM problem involving all flow rates to a simpler problem involving only the aggregate rates for each source-destination pair. The optimal aggregate flows are then apportioned among the constituent flows of each pair. This apportionment is simple for the case of $\alpha$-fair utility functions. We also show how the decomposition can be implemented with the alternating direction method of multipliers (ADMM) algorithm.