Hardness of Virtual Network Embedding with Replica Selection

Efficient embedding virtual clusters in physical network is a challenging problem. In this paper we consider a scenario where physical network has a structure of a balanced tree. This assumption is justified by many realworld implementations of datacenters. We consider an extension to virtual cluster embedding by introducing replication among data chunks. In many real-world applications, data is stored in distributed and redundant way. This assumption introduces additional hardness in deciding what replica to process. By reduction from classical NP-complete problem of Boolean Satisfiability, we show limits of optimality of embedding. Our result holds even in trees of edge height bounded by three. Also, we show that limiting replication factor to two replicas per chunk type does not make the problem simpler.