Counting Euler Tours in Undirected Bounded Treewidth Graphs

TL;DR

This paper demonstrates that counting Euler tours in undirected bounded tree-width graphs is computationally feasible in parallel, contrasting with its intractability in general graphs, by leveraging dynamic programming and parallel algorithms.

Contribution

It introduces a parallel algorithm for counting Euler tours in bounded tree-width graphs, extending techniques from Hamiltonian path counting and applying polynomial interpolation for uniform complexity bounds.

Findings

Counting Euler tours in bounded tree-width graphs is in ^1.

Parallel algorithms for counting longest paths and bipartite perfect matchings in bounded clique-width graphs.

Establishes ^1 complexity bounds using polynomial interpolation.

Abstract

We show that counting Euler tours in undirected bounded tree-width graphs is tractable even in parallel - by proving a $\#SAC^1$ upper bound. This is in stark contrast to #P-completeness of the same problem in general graphs. Our main technical contribution is to show how (an instance of) dynamic programming on bounded \emph{clique-width} graphs can be performed efficiently in parallel. Thus we show that the sequential result of Espelage, Gurski and Wanke for efficiently computing Hamiltonian paths in bounded clique-width graphs can be adapted in the parallel setting to count the number of Hamiltonian paths which in turn is a tool for counting the number of Euler tours in bounded tree-width graphs. Our technique also yields parallel algorithms for counting longest paths and bipartite perfect matchings in bounded-clique width graphs. While establishing that counting Euler tours in bounded tree-width graphs can be computed by non-uniform monotone arithmetic circuits of polynomial degree (which characterize $\#SAC^1$) is relatively easy, establishing a uniform $\#SAC^1$ bound needs a careful use of polynomial interpolation.