  The advent of network coding presents promising opportunities in many areas
of communication and networking. It has been recently shown that network coding
technique can significantly increase the overall throughput of wireless
networks by taking advantage of their broadcast nature. In wireless networks,
each transmitted packet is broadcasted within a certain area and can be
overheard by the neighboring nodes. When a node needs to transmit packets, it
employs the opportunistic coding approach that uses the knowledge of what the
node's neighbors have heard in order to reduce the number of transmissions.
With this approach, each transmitted packet is a linear combination of the
original packets over a certain finite field.
  In this paper, we focus on the fundamental problem of finding the optimal
encoding for the broadcasted packets that minimizes the overall number of
transmissions. We show that this problem is NP-complete over GF(2) and
establish several fundamental properties of the optimal solution. We also
propose a simple heuristic solution for the problem based on graph coloring and
present some empirical results for random settings.
