A Note on Property Testing of the Binary Rank
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Let M be a n× m (0,1)-matrix. We define the s-binary rank, br_s(M), of M to be the minimal integer d such that there are d monochromatic rectangles that cover all the 1-entries in the matrix, and each 1-entry is covered by at most s rectangles. When s=1, this is the binary rank, br(M), known from the literature. Let R(M) and C(M) be the set of rows and columns of M, respectively. We use the result of Sgall (Comb. 1999) to prove that if M has s-binary rank at most d, then |R(M)|· |C(M)|≤d≤ s2^d where d≤ s=∑_i=0^sd i. This bound is tight; that is, there exists a matrix M' of s-binary rank d such that |R(M')|· |C(M')|= d≤ s2^d. Using this result, we give a new one-sided adaptive and non-adaptive testers for (0,1)-matrices of s-binary rank at most d (and exactly d) that makes Õ(d≤ s2^d/ϵ) and Õ(d≤ s2^d/ϵ^2) queries, respectively. For a fixed s, this improves the query complexity of the tester of Parnas et al. (Theory Comput. Syst. 2021) by a factor of Θ̃(2^d).
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