Network Coding with a Cost Criterion

We consider applying network coding in settings where there is a cost associated with network use. We show that, while minimum-cost multicast problems without network coding are very difficult except in the special cases of unicast and broadcast, finding minimum-cost subgraphs for single multicast connections with network coding can be posed as a linear optimization problem. In particular, we apply our approach to the problem of minimum-energy multicast in wireless networks with omnidirectional antennas and show that it can be handled by a linear optimization problem when network coding is used. For the case of multiple multicast connections, we give a partial solution: We specify a linear optimization problem that yields a solution of no greater cost than any solution without network coding and that we suspect can potentially be substantially better.