On the Partition Bound for Undirected Unicast Network Information Capacity

TL;DR

This paper investigates the partition bound in undirected unicast networks, proving its computational complexity, characterizing optimal routing schemes for certain networks, and exploring its relation to the network coding conjecture.

Contribution

It establishes the NP-completeness of computing the partition bound, provides optimal routing schemes for specific network classes, and presents a counterexample related to the network coding conjecture.

Findings

Decision problem for partition bound is NP-complete.

Optimal routing schemes are characterized for two network classes.

Existence of a network where the partition bound is tight and not in the known class.

Abstract

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.