5077 solvable over A. But, by Theorem III.10, network N6 is not solvable over any alphabet, so neither is the reverse of N 7 . Hence, N 7 is not reversible. Also, by Theorem III.10, the reverse of N 6 is solvable, and thus N7 is solvable, by Theorem III.11. We have demonstrated a multiple-unicast network which is solvable, but whose reverse is not solvable. However, we note that the network and its reverse each have coding capacity equal to one. In particular, this implies that the reverse network is asymptotically solvable.  Sandeep Bhadra, Student Member, IEEE, Sanjay Shakkottai, Member, IEEE, and Piyush Gupta, Member, IEEE Abstract-The single-source min-cost multicast problem, which can be framed as a convex optimization problem with the use of network codes and convex increasing edge costs is considered. A decentralized approach to this problem is presented by Lun, Ratnakar et al. for the case where all users cooperate to reach the global minimum. Further, the cost for the scenario where each of the multicast receivers greedily routes its flows is analyzed and the existence of a Nash equilibrium is proved. An allocation rule by which edge cost at each edge is allocated to flows through that edge is presented. We prove that under our pricing rule, the flow cost at user equilibrium is the same as the min-cost. This leads to the construction of a selfish flow-steering algorithm for each receiver, which is also globally optimal. Further, the algorithm is extended for completely distributed flow adaptation at nodes in the network to achieve globally minimal cost in steady state. Analogous results are also presented for the case of multiple multicast sessions.
The work that is closest to our analysis is by Bhadra et al. REF that considers the problem of minimum cost multicast with network coding in the case of single source and many selfish terminals.
Min-Cost Selfish Multicast With Network Coding
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Computer Science