Property Testing of the Boolean and Binary Rank

Abstract

We present algorithms for testing if a (0,1)-matrix M has Boolean/binary rank at most d, or is 𝜖-far from having Boolean/binary rank at most d (i.e., at least an 𝜖-fraction of the entries in M must be modified so that it has rank at most d). For the Boolean rank we present a non-adaptive testing algorithm whose query complexity is \(\tilde {O}\left (d^{4}/ \epsilon ^{6}\right )\). For the binary rank we present a non-adaptive testing algorithm whose query complexity is O(22d/𝜖2), and an adaptive testing algorithm whose query complexity is O(22d/𝜖). All algorithms are 1-sided error algorithms that always accept M if it has Boolean/binary rank at most d, and reject with probability at least 2/3 if M is 𝜖-far from having Boolean/binary rank at most d.