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We consider a setting in which a sender wishes to broadcast a block of K data packets to a set of wireless receivers, where each of the receivers already has a subset of the data packets available to it (e.g., from prior transmissions) and wants to obtain the rest of the packets in the block. Our goal is to find a linear network coding scheme that yields the minimum average packet decoding delay (APDD), i.e., the average time it takes for a receiver to decode a data packet. Our contributions can be summarized as follows. First, we prove that this problem is NP-hard by presenting a reduction from the hypergraph coloring problem. Next, we show that the random linear network coding (RLNC) technique provides an approximate solution to this problem with approximation ratio of 2 with high probability. Next, we present a methodology for designing specialized approximation algorithms for this problem that outperform RLNC solutions while maintaining the same throughput. In a special case of practical interest in which each receiver wants a small number of packets, our solution can achieve an approximation ratio of 4-2/K 3. Finally, we conduct an experimental study that demonstrates the advantages of the presented methodology.

On minimizing the average packet decoding delay in wireless network coded broadcast