# An Approximation Algorithm for Path Computation and Function Placement in SDNs

## Abstract

We consider the task of embedding multiple service requests in Software-Defined Networks (SDNs), i.e. computing (combined) mappings of network functions on physical nodes and finding routes to connect the mapped network functions. A single service request may either be fully embedded or be rejected. The objective is to maximize the sum of benefits of the served requests, while the solution must abide node and edge capacities.

We follow the framework suggested by Even *et al.* [5] for the specification of the network functions and routing of requests via processing-and-routing graphs (PR-graphs): a request is represented as a directed acyclic graph with the nodes representing network functions. Additionally, a unique source and a unique sink node are given for each request, such that any source-sink path represents a feasible chain of network functions to realize the service. This allows for example to choose between different realizations of the same network function. Requests are attributed with a global demand (e.g. specified in terms of bandwidth) and a benefit.

Our main result is a randomized approximation algorithm for path computation and function placement with the following guarantee. Let *m* denote the number of links in the substrate network, \(\varepsilon \) denote a parameter such that \(0< \varepsilon <1\), and \(\mathsf {opt}^*\) denote the maximum benefit that can be attained by a fractional solution (one in which requests may be partly served and flow may be split along multiple paths). Let \(c_{\min }\) denote the minimum edge capacity, let \(d_{\max }\) denote the maximum demand, and let \(b_{\max }\) denote the maximum benefit of a request. Let \(\varDelta _{\max }\) denote an upper bound on the number of processing stages a request undergoes. If \(c_{\min }/(\varDelta _{\max }\cdot d_{\max })=\varOmega ((\log m)/\varepsilon ^2)\), then with probability at least \(1-\frac{1}{m}-\textit{exp}(-\varOmega (\varepsilon ^2\cdot \mathsf {opt}^*/(b_{\max }\cdot d_{\max })))\), the algorithm computes a \((1-\varepsilon )\)-approximate solution.

## Notes

### Acknowledgment

This research was supported by the EU project UNIFY FP7-IP-619609 as well as by the German BMBF Software Campus grant 01IS1205. Stefan Schmid is supported by the Aalborg University’s inter-faculty Talent Program.

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