On the complexity of Boolean matrix ranks

We construct a reduction which proves that the fooling set number and the determinantal rank of a Boolean matrix are NP-hard to compute. This note is devoted to the functions of determinantal rank and fooling set number, which are receiving attention in different applications, see [1, 3] and references therein. The purpose of this note is to give an NP-completeness proof for those functions, thus answering questions asked in [1, Chapter 6] and [3]. A Boolean matrix A ∈ {0, 1} is called sign-nonsingular if the condition A1,τ(1)=. . .=An,τ(n)=1 holds for some permutation τ on {1, . . . , n} and for no permutation of parity different from that of τ . The determinantal rank of a matrix is the size of its largest sign-nonsingular square submatrix. Entries (i1, j1), . . . , (in, jn) of a Boolean matrix B are called a fooling set if Bis,jt = Bit,js = 1 holds exactly when t = s. The fooling set number, equal to the cardinality of the largest fooling set, is clearly an upper bound for the determinantal rank. Consider a directed graph G = (V,E) in which (u, v) ∈ E implies (v, u) ∈ E and construct the matrix A = A(G) with rows and columns indexed by V ∪ E as follows. Given x, y ∈ V ∪ E, we set Axy = 1 when x = y or (x, y) ∈ E, or else y = (x, u) or x = (u, y), for some vertex u; otherwise, we set Axy = 0. If U is an independent set in G, then by our construction the submatrix of A with rows and columns indexed by U ∪ E is sign-nonsingular. On the other hand, consider a subset S ⊂ (V ∪E)×(V ∪E) such that |S| = |E|+h. If Aij with (i, j) running over S is a fooling set, then the intersection of S with {e, u} × {e, v} has cardinality 0 or 1, for every e = (u, v) ∈ E. So S contains at least h elements of the form (w,w) with w ∈ V , and those vertices w form an independent set of cardinality h in G. The fooling set number and determinantal rank of A(G) are both therefore equal to |U |+ |E|, where U ′ is the largest independent set in G. Checking whether a graph has an independent set of size exceeding k is a classical NP-complete problem, so it is NP-hard to decide whether or not the determinantal rank or fooling set number of a matrix exceed k. Those problems are in fact NPcomplete — a polynomial algorithm for sign-singularity is a deep result [2] while yet a straightforward algorithm recognizing fooling sets is polynomial.