The computational complexity of some problems of linear algebra

Abstract

In this paper we consider the computational complexity of some problems dealing with matrix rank. Let E, S be subsets of a commutative ring R. Let x ₁, x ₂, ..., x _t be variables. Given a matrix M= M(x ₁, x ₂, ..., x _t ) with entries chosen from E ∪ {x ₁, x ₂, ..., x _t }, we want to determine

$$\max rank_S (M) = \mathop {max}\limits_{(a_1 ,a_2 ,...a_t ) \in S^t } rank M(a_1 ,a_2 ,...a_t )$$

and

$$\min rank_S (M) = \mathop {min}\limits_{(a_1 ,a_2 ,...a_t ) \in S^t } rank M(a_1 ,a_2 ,...a_t ).$$

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.

Supported in part by grants from NSERC Canada.

Supported in part by the ESPRIT Long Term Research Programme of the EU under project number 20244 (ALCOM-IT) and by BRICS, Basic Research in Computer Science, Centre of the Danish National Research Foundation.