Tree-width and Logspace: Determinants and Counting Euler Tours

TL;DR

This paper demonstrates that several complex graph properties, including determinants, Euler tours, and spanning trees, can be computed in logarithmic space for graphs with bounded tree-width, extending the class of efficiently computable problems.

Contribution

It proves that computing the determinant, counting Euler tours, and related linear algebraic properties are in L for bounded tree-width graphs, extending known results beyond MSO properties.

Findings

Determinant of adjacency matrix is L-complete for bounded tree-width graphs.

Counting Euler tours and spanning arborescences is in L for bounded tree-width digraphs.

Polynomial time algorithms for counting Euler tours in bounded tree-width graphs are established.

Abstract

Motivated by the recent result of [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 prove that it is L-complete. It is important to notice that the determinant is neither an MSO-property nor counts the number of solutions of an MSO-predicate. We extend this technique to count the number of spanning arborescences and directed Euler tours in bounded tree-width digraphs, and further to counting the number of spanning trees and the number of Euler tours in undirected graphs, all in L. Notice that undirected Euler tours are not known to be MSO-expressible and the corresponding counting problem is in fact #P-hard for general graphs. Counting undirected Euler tours in bounded tree-width graphs was not known to be polynomial time computable till very recently Chebolu et al [CCM13] gave a polynomial time algorithm for this problem (concurrently and independently of this work). Finally, we also show some linear algebraic extensions of the determinant algorithm to show how to compute the charcteristic polynomial and trace of the powers of a bounded tree-width graph in L.