Abstract. Motivated by a recent result of Elberfeld, Jakoby and Tantau [EJT10] showing that 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 MSO-property nor counts the number of solutions of an 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 L-hardness of the problems of computing the determinant, and of powering a bounded tree-width matrix. We also show the GapL-hardness of Iterated Matrix Multiplication where each matrix has bounded tree-width.
For directed (resp., undirected) graphs of bounded tree width, the number of Eulerian circuits can be computed in logspace REF (resp.
Bounded Treewidth and Space-Efficient Linear Algebra
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Mathematics Computer Science