On the capacity of sum-networks

In this paper, we consider sum-networks, which is a class of directed acyclic networks where each terminal in the network requires sum of all the sources. In the literature, all reported sum-networks are shown to have only certain rational valued coding capacities. For example, it has been shown that a sum-network, having 3 sources and 3 terminals, has coding capacity either 0, 2/3 or ≥ 1. It is an open problem whether every positive rational number is the coding capacity of some sum-network. In this paper, for every positive rational number k/n, we show the existence of a sum-network which has coding capacity equal to k/n. The constructed sum-network also demonstrate that the gap between min-cut bound and coding capacity can be unbounded for sum-networks.