Orthogonal representations over finite fields and the chromatic number of graphs

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Weighted low rank approximations with provable guarantees (2016, background): “org/wiki/Low-rank_ approximation#Weighted_low-rank_approximation_ problems mation in which case all weights are 0 or 1, which we discuss more below, is also known to be NP-hard [46, 29].”
Low-rank matrix completion using alternating minimization (2012, background): “Without any extra conditions, both matrix sensing and matrix completion are ill-posed problems, with potentially multiple low-rank solutions, and are in general NP hard [20, 21].”
How hard is the tensor rank (2016, background): “Other related completion problems whose complexity has been known are the Euclidean distance completion ([25]), minimal rank sign pattern completion ([2]), and the problem of minimizing the rank of matrices ﬁtting a given graph ([31]).”
Index Coding With Side Information (2006, background): “Their main thrust was proving tight bounds on the length of the codes in the quantum setting, where inputs can be encoded by qubits rather than classical bits; their result applied to the classical setting is a special case of our Theorem 3 for the case when is the empty graph.”