A fast matrix completion method for index coding

We investigate the problem of index coding, where a sender transmits distinct packets over a shared link to multiple users with side information. The aim is to find an encoding scheme (linear combinations) to minimize the number of transmitted packets, while providing each user with sufficient amount of data for the recovery of the desired parts. It has been shown that finding the optimal linear index code is equivalent to a matrix completion problem, where the observed elements of the matrix indicate the side information available for the users. This modeling results in an incomplete square matrix with all ones on the main diagonal (and some other parts), which needs to be completed with minimum rank. Unfortunately, this is a case in which conventional matrix completion techniques based on nuclear-norm minimization are proved to fail [Huang, Rouayheb 2015]. Instead, an alternating projection (the AP algorithm) method is proposed in [Huang, Rouayheb 2015]. In this paper, in addition to proving the convergence of the AP algorithm under certain conditions, we introduce a modification which considerably improves the run time of the method.