Sum-networks: System of polynomial equations, unachievability of coding capacity, reversibility, insufficiency of linear network coding

We consider a directed acyclic network where all the terminals demand the sum of the symbols generated at all the sources. We call such a network as a sum-network. We show that there exists a solvably (linear solvably) equivalent sum-network for any multiple-unicast network (and more generally, for any acyclic directed network where each terminal node demands a subset of the symbols generated at all the sources). We also show that there exists a linear solvably equivalent multiple-unicast network for every sum-network. As a consequence, some of the known results for multipleunicast networks (or acyclic directed network where each terminal node demands a subset of the symbols generated at all the sources) are applicable for the sum-networks also. Specifically, we show that for any set of polynomials having integer coefficients, there exists a sum-network which is scalar linear solvable over a finite field F if and only if the polynomials have a common root in F . We show the insufficiency of linear network coding for sumnetworks and unachievability of the network coding capacity of sum-networks. It is shown that there exists a solvable sumnetwork whose reverse network is not solvable. We also show that communicating the sum of symbols and communicating any linear function is solvably equivalent for any alphabet field.