Exact counting of Euler Tours for generalized series-parallel graphs

TL;DR

This paper presents the first polynomial-time algorithms for exactly counting and sampling Euler Tours in generalized series-parallel graphs, including all outerplanar graphs, using a novel dynamic programming approach.

Contribution

It introduces a simple polynomial-time algorithm for counting and sampling Euler Tours in generalized series-parallel graphs, a class that includes all outerplanar graphs.

Findings

Counting and sampling Euler Tours is feasible in polynomial time for generalized series-parallel graphs.

The algorithms run in $O(m riangle^3)$ time and use $O(m riangle^2 ext{log} riangle)$ bits.

This is the first known polynomial-time solution for these problems in any graph class.

Abstract

We give a simple polynomial-time algorithm to exactly count the number of Euler Tours (ETs) of any Eulerian generalized series-parallel graph, and show how to adapt this algorithm to exactly sample a random ET of the given generalized series-parallel graph. Note that the class of generalized seriesparallel graphs includes all outerplanar graphs. We can perform the counting in time $O(m\Delta^3)$, where $\Delta$ is the maximum degree of the graph with $m$ edges. We use $O(m\Delta^2 \log \Delta)$ bits to store intermediate values during our computations. To date, these are the first known polynomial-time algorithms to count or sample ETs of any class of graphs; there are no other known polynomial-time algorithms to even approximately count or sample ETs of any other class of graphs. The problem of counting ETs is known to be $#P$-complete for general graphs (Brightwell and Winkler, 2005 [3]) and also for planar graphs (Creed, 2009 [4]).