Bounded Treewidth and Space-Efficient Linear Algebra

TL;DR

This paper proves that computing the determinant of a bounded tree-width graph's adjacency matrix is in Logspace, enabling efficient algorithms for related linear algebra problems on such graphs.

Contribution

It introduces Logspace algorithms for determinant and related linear algebra computations on bounded tree-width graphs, and establishes hardness results for these problems.

Findings

Determinant computation for bounded tree-width graphs is in Logspace.

Logspace algorithms are developed for counting spanning arborescences and Euler tours.

Hardness results show these problems are L-hard or GapL-hard.

Abstract

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.