Algorithms for the Maximum Eulerian Cycle Decomposition Problem

TL;DR

This paper introduces algorithms for decomposing Eulerian graphs into maximum sets of edge-disjoint cycles, including an ILP-based approach, a greedy heuristic, and a partial ILP heuristic, with experimental comparisons.

Contribution

It presents a novel ILP-based algorithm for the maximum Eulerian cycle decomposition problem and compares it with heuristics through extensive experiments.

Findings

ILP-based heuristic outperforms other methods in solution quality

The algorithms are tested on various Eulerian graphs with different sizes

Experimental results demonstrate the effectiveness of the ILP approach

Abstract

Given an Eulerian graph G, in the Maximum Eulerian Cycle Decomposition problem, we are interested in finding a collection of edge-disjoint cycles {E_1, E_2, ..., E_k} in G such that all edges of G are in exactly one cycle and k is maximum. We present an algorithm to solve the pricing problem of a column generation Integer Linear Programming (ILP) model introduced by Lancia and Serafini (2016). Furthermore, we propose a greedy heuristic, which searches for minimum size cycles starting from a random vertex, and a heuristic based on partially solving the ILP model. We performed tests comparing the three approaches in relation to the quality of solutions and execution time, using distinct sets of Eulerian graphs, each set grouping graphs with different numbers of vertices and edges. Our experimental results show that the ILP based heuristic outperforms the other methods.