Network Slicing with Splittable Flows is Hard.

Allocating resources to network slices can be achieved by means of solving virtual network embedding problems, whereby virtual nodes are used to reserve computing resources on cloud nodes, and virtual links are used to reserve bandwidth resources on network paths. Since the associated optimization problem is also NP-hard to approximate, in this paper we focus on a natural simplified setting of interest: the case where the tunnels can be embedded with splittable flows. For this problem, we provide a simple proof that it is NP-hard by a reduction from the 3-SAT problem. Further, using the idea of the multipartite graph, we propose a poly-time heuristic for the loose capacity constraint case, based on linear relaxation and randomized rounding. This heuristic is shown to have small optimality gaps in extensive simulations.