The Computational Complexity of Some Problems of Linear Algebra (Extended Abstract)

We consider the computational complexity of some problems dealing with matrix rank. Let E ; S be subsets of a commutative ring R. we want to determine maxrank S (M) = max (a 1 ;a 2 ;:::;at)2S t rank M (a 1 ; a 2 ; : : : a t) and minrank S (M) = min (a 1 ;a 2 ;:::;at)2S t rank M (a 1 ; a 2 ; : : : a t): There are also variants of these problems that specify more about the structure of M , or instead of asking for the minimum or maximum rank, ask if there is some substitution of the variables that makes the matrix invertible or noninvertible. Depending on E ; S , and on which variant is studied, the complexity of these problems can range from polynomial-time solvable to random polynomial-time solvable to NP-complete to PSPACE-solvable to unsolvable.