Abstract—The question of under what condition some side information for index coding can be removed with-out affecting the capacity region is studied, which was originally posed by Tahmasbi, Shahrasbi, and Gohari. To answer this question, the notion of unicycle for the side information graph is introduced and it is shown that any edge that belongs to a unicycle is critical, namely, it cannot be removed without reducing the capacity region. Although this sufficient condition for criticality is not necessary in general, a partial converse is established, which elucidates the connection between the notion of unicycle and the maximal acylic induced subgraph outer bound on the capacity region by Bar-Yossef, Birk, Jayram, and Kol. I.

Abstract—Index codes reduce the number of bits broadcast by a wireless transmitter to a number of receivers with different demands and with side information. It is known that the problem of finding optimal linear index codes is NP-hard. We investigate the performance of different heuristics based on rank minimization and matrix completion methods, such as alternating projections and alternating minimization, for constructing linear index codes over the reals. As a summary of our results, the alternating projections method gives the best results in terms of minimizing the number of broadcast bits and convergence rate and leads to up to 13 % savings in average communication cost compared to graph coloring algorithms studied in the literature. Moreover, we describe how the proposed methods can be used to construct linear network codes for non-multicast networks. Our computer code is available online. I.

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