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

A directed acyclic network is considered where all the terminals demand the sum of the symbols generatedat all the sources. We call such a network as a sum-network. It is shown that there exists a solvably (and linearsolvably) equivalent sum-network for any multiple-unicast network (and more generally, for any acyclic directednetwork where each terminal node demands a subset of the symbols generated at all the sources). It is also shownthat there exists a linear solvably equivalent multiple-unicast network for every sum-network. As a consequence,many known results for multiple-unicast networks also hold for sum-networks. Specifically, it is shown that for anyset of polynomials having integer coefficients, there exists a sum-network which is scalar linear solvable over afinite field F if and only if the polynomials have a common root in F . Similarly, the insufficiency of linear networkcoding and unachievability of the network coding capacity is proved for sum-networks. It is shown that there existsa solvable sum-network whose reverse network is not solvable. On the other hand, a sum-network and its reversenetwork are shown to be solvably equivalent under fractional vector linear network coding.