FPT Algorithms Exploiting Carving Decomposition for Eulerian Orientations and Ice-Type Models

Abstract

An Eulerian orientation of an undirected graph is an orientation of edges such that, for each vertex, both the indegree and the outdegree are the same. Eulerian orientations are important in a variety of fields. In statistical physics, the partition function of the so-called ice model, which is the special case of the ice-type model, is related to the number of Eulerian orientations of a 4-regular graph, which is the value of its Tutte polynomial at the point $(0,-2)$. The problem of counting the number of Eulerian orientations in a 4-regular graph is #P-complete, and yet there is an FPT (Fixed Parameter Tractable) algorithm for it with respect to the tree-width of the graph.

This paper presents two FPT algorithms based on a carving decomposition. One of them counts the number of Eulerian orientations for a general graph in $O(k \cdot (2 \sqrt{2})^k \cdot n)$ time and $O(2^k \cdot n)$ memory consumption, and the other calculates the partition function of a general ice-type model for a 4-regular graph in $O(k \cdot (2 \sqrt{2})^k \cdot n)$ time and $O(2^k \cdot n + (2 \sqrt{2})^k)$ memory consumption where, for an input graph, k is the carving-width and n is the size of the vertex set.