# Approximate and Incremental Network Function Placement

## TL;DR

This paper introduces a new approximation algorithm for the incremental placement of network middleboxes, ensuring capacity and route constraints are met, with proven optimal bounds and extensions to weighted communication scenarios.

## Contribution

It provides a rigorous proof of submodularity for the network function placement problem and develops a deterministic, incremental deployment algorithm with optimal approximation guarantees.

## Key findings

- The algorithm effectively handles incremental middlebox deployment without relocating existing middleboxes.
- The approximation bound achieved is proven to be optimal unless P=NP.
- A new LP relaxation and randomized rounding approach are proposed for weighted communication scenarios.

## Abstract

The virtualization and softwarization of modern computer networks introduces interesting new opportunities for a more flexible placement of network functions and middleboxes (firewalls, proxies, traffic optimizers, virtual switches, etc.). This paper studies approximation algorithms for the incremental deployment of a minimum number of middleboxes at optimal locations, such that capacity constraints at the middleboxes and length constraints on the communication routes are respected. Our main contribution is a new, purely combinatorial and rigorous proof for the submodularity of the function maximizing the number of communication requests that can be served by a given set of middleboxes. Our proof allows us to devise a deterministic approximation algorithm which uses an augmenting path approach to compute the submodular function. This algorithm does not require any changes to the locations of existing middleboxes or the preemption of previously served communication pairs when additional middleboxes are deployed, previously accepted communication pairs just can be handed over to another middlebox. It is hence particularly attractive for incremental deployments.We prove that the achieved polynomial-time approximation bound is optimal, unless P = NP. This paper also initiates the study of a weighted problem variant, in which entire groups of nodes need to communicate via a middlebox (e.g., a multiplexer or a shared object), possibly at different rates. We present an LP relaxation and randomized rounding algorithm for this problem, leveraging an interesting connection to scheduling.