One of the important unsolved problems in information theory is the conjecture that the undirected multiple-unicast network information capacity is the same as the routing capacity. This conjecture is verified only for a handful of networks and network classes. Moreover, only two explicit upper bounds on information capacity are known for general undirected networks: the sparsest cut bound and the linear programming bound. In this paper, we present an information-theoretic upper bound, called thepartition bound, on the capacity of general undirected multiple-unicast networks. We show that a decision version problem of computing the bound is NP-complete. We present two classes of undirected multiple-unicast networks such that the partition bound is achievable by routing. Thus, the conjecture is proved for these classes of networks. Recently, the conjecture was proved for a new class of networks defined by properties relating to cut-set and source-sink paths. We show the existence of a network outside of this new class of networks such that the partition bound is achievable by routing.

One of the important unsolved problems in information theory is the conjecture that network coding has no rate benefit over routing in undirected unicast networks. Three known bounds on the symmetric rate in undirected unicast information networks are the sparsest cut, the LP bound and the partition bound. In this paper, we present three results on the partition bound. We show that the decision version problem of computing the partition bound is NP-complete. We give complete proofs of optimal routing schemes for two classes of networks that attain the partition bound. Recently, the conjecture was proved for a new class of networks and it was shown that all the network instances for which the conjecture is proved previously are elements of this class. We show the existence of a network for which the partition bound is tight, achievable by routing and is not an element of this new class of networks.

In this paper, we present a new technique to obtain upper bounds on undirected unicast network information capacity. Using this technique, we characterize an upper bound, called partition bound, on the symmetric rate of information flow in undirected unicast networks and give an algorithm to compute it. Two classes of networks are presented for which the bound is tight and the capacity is achievable by routing thus confirming the undirected unicast conjecture for these classes of networks. We also show that the bound can be loose in general and present an approach to tighten it.