The Computational Complexity of Some Problems of Linear Algebra

We consider the computational complexity of some problems dealing with matrix rank. Let E, S be subsets of a commutative ring R. Let x1, x2, ?, xt be variables. Given a matrix M=M(x1, x2, ?, xt) with entries chosen from E?{x1, x2, ?, xt}, we want to determine maxrankS(M)=max(a1, a2, ?, at)?St rank M(a1, a2, ?, at) and minrankS(M)=min(a1, a2, ?, at)?St rank M(a1, a2, ?, at). 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, they ask if there is some substitution of the variables that makes the matrix invertible or noninvertible. Depending on E, S, and 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. An approximation version of the minrank problem is shown to be MAXSNP-hard.

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