[Paper Review] Exact counting of Euler Tours for Graphs of Bounded Treewidth

Prasad Chebolu, Mary Cryan+1 more|arXiv (Cornell University)|Oct 1, 2013

Advanced Graph Theory Research3 references3 citations

TL;DR

This paper presents a polynomial-time algorithm to exactly count Euler tours in undirected Eulerian graphs of bounded treewidth, leveraging dynamic programming on a nice tree decomposition. The key contribution is extending prior results from series-parallel graphs (treewidth 2) to all graphs with fixed treewidth, resolving a long-standing open problem in counting complexity for this graph class.

ABSTRACT

In this paper we give a simple polynomial-time algorithm to exactly count the number of Euler Tours (ETs) of any Eulerian graph of bounded treewidth. The problems of counting ETs are known to be #P-complete for general graphs (Brightwell and Winkler, (Brightwell and Winkler, 2005). To date, no polynomial-time algorithm for counting Euler tours of any class of graphs is known except for the very special case of series-parallel graphs (which have treewidth 2).

Motivation & Objective

• To develop an efficient exact counting algorithm for Euler tours in undirected Eulerian graphs with bounded treewidth.
• To extend the known polynomial-time result for series-parallel graphs (treewidth 2) to all graphs of fixed treewidth.
• To address the computational hardness of Euler tour counting in general graphs, which is known to be #P-complete.
• To provide a constructive method that leverages tree decomposition and dynamic programming on the decomposition.

Proposed method

• The algorithm uses a nice tree decomposition of the input graph, which is a standard technique for bounded treewidth graphs.
• It maintains dynamic programming tables that track charge vectors and root vectors for partial forests in each subtree.
• For join nodes in the tree decomposition, the algorithm combines solutions from left and right subtrees using combinatorial rules that preserve consistency of root assignments and charge propagation.
• The algorithm computes valid combinations by checking conditions on root vectors to ensure acyclicity and proper connectivity in the resulting forests.
• It uses a recursive bottom-up traversal of the tree decomposition, initializing tables at leaves and combining them at internal nodes.
• The method ensures correctness by enforcing constraints that correspond to valid Eulerian orientations and consistent edge traversals.

Experimental results

Research questions

• RQ1 Can the number of Euler tours be counted in polynomial time for graphs of bounded treewidth, despite the #P-completeness of the problem in general?
• RQ2 Is there a dynamic programming approach on tree decompositions that can efficiently count Euler tours by tracking structural properties like charge and root vectors?
• RQ3 How can the relationship between Euler tours and in-arborescences in directed Eulerian graphs be exploited for counting in undirected graphs of bounded treewidth?
• RQ4 Can the algorithm be extended to compute other graph polynomials, such as the Martin polynomial, for bounded treewidth graphs?

Key findings

• The paper presents a polynomial-time algorithm for counting Euler tours in any Eulerian graph of bounded treewidth, solving an open problem in counting complexity.
• The algorithm is based on dynamic programming over a nice tree decomposition, with state tables tracking charge and root vectors for partial forests.
• The method correctly handles join nodes by combining solutions from subtrees under constraints ensuring acyclicity and consistent root assignments.
• The approach generalizes prior work on series-parallel graphs (treewidth 2) to all graphs of fixed treewidth d.
• The algorithm’s correctness is grounded in a precise characterization of how Euler tours relate to in-arborescences and forest structures in the decomposition.
• The work provides a foundation for potentially computing other graph polynomials, such as the Martin polynomial, for bounded treewidth graphs.