Bounded Treewidth and Space-Efficient Linear Algebra

Motivated by a recent result of Elberfeld, Jakoby and Tantau showing that $\mathsf{MSO}$ properties are Logspace computable on graphs of bounded tree-width, we consider the complexity of computing the determinant of the adjacency matrix of a bounded tree-width graph and as our main result prove that it is in Logspace. It is important to notice that the determinant is neither an $\mathsf{MSO}$-property nor counts the number of solutions of an $\mathsf{MSO}$-predicate. This technique yields Logspace algorithms for counting the number of spanning arborescences and directed Euler tours in bounded tree-width digraphs. We demonstrate some linear algebraic applications of the determinant algorithm by describing Logspace procedures for the characteristic polynomial, the powers of a weighted bounded tree-width graph and feasibility of a system of linear equations where the underlying bipartite graph has bounded tree-width. Finally, we complement our upper bounds by proving $\mathsf{L}$-hardness of the problems of computing the determinant, and of powering a bounded tree-width matrix. We also show the $\mathsf{GapL}$-hardness of Iterated Matrix Multiplication where each matrix has bounded tree-width.