We consider the task of computing (combined) function mapping and routing for requests in Software-Defined Networks (SDNs). Function mapping refers to the assignment of nodes in the substrate network to various processing stages that requests must undergo. Routing refers to the assignment of a path in the substrate network that begins in a source node of the request, traverses the nodes that are assigned functions for this request, and ends in a destination of the request. The algorithm either rejects a request or completely serves a request, and its goal is to maximize the sum of the benefits of the served requests. The solution must abide edge and vertex capacities. We follow the framework suggested by Even for the specification of the processing requirements and routing of requests via processing-and-routing graphs (PR-graphs). In this framework, each request has a demand, a benefit, and PR-graph. 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, $eps$ denote a parameter such that $0< eps <1$, and $opt_f$ 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 $cmin$ denote the minimum edge capacity, and let $dmax$ denote the maximum demand. Let $Deltamax$ denote an upper bound on the number of processing stages a request undergoes. If $cmin/(Deltamaxcdotdmax)=Omega((log m)/eps^2)$, then with probability at least $1-frac{1}{m}-textit{exp}(-Omega(eps^2cdot opt_f /(bmax cdot dmax)))$, the algorithm computes a $(1-eps)$-approximate solution.
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