On Making Directed Graphs Eulerian

TL;DR

This paper investigates the Eulerian extension problem in directed graphs, analyzing its parameterized complexity with respect to graph components and degree imbalances, and proposes algorithms and complexity bounds for solving it.

Contribution

It introduces a new parameterized analysis of the Eulerian extension problem, including a polynomial-time algorithm with a specific exponential dependence and a matching formulation approach.

Findings

Algorithm with running time polynomial except for 4^(c log(bc^2)) term.

Matching formulation of EE as a potential solving tool.

Polynomial-time preprocessing cannot reduce instance size polynomially based on parameters unless coNP ⊆ NP/poly.

Abstract

A directed graph is called Eulerian, if it contains a tour that traverses every arc in the graph exactly once. We study the problem of Eulerian extension (EE) where a directed multigraph G and a weight function is given and it is asked whether G can be made Eulerian by adding arcs whose total weight does not exceed a given threshold. This problem is motivated through applications in vehicle routing and flowshop scheduling. However, EE is NP-hard and thus we use the parameterized complexity framework to analyze it. In parameterized complexity, the running time of algorithms is considered not only with respect to input length, but also with respect to other properties of the input - called "parameters". Dorn et. al. proved that EE can be solved in O(4^k n^4) time, where k denotes the parameter "number of arcs that have to be added". In this thesis, we analyze EE with respect to the (smaller) parameters "number c of connected components in the input graph" and "sum b over indeg(v) - outdeg(v) for all vertices v in the input graph where this value is positive". We prove that there is an algorithm for EE whose running time is polynomial except for the term 4^(c log(bc^2)). To obtain this result, we make several observations about the sets of arcs that have to be added to the input graph in order to make it Eulerian. We build upon these observations to restate EE in a matching context. This matching formulation of EE might be an important tool to solve the question of whether EE can be solved within running time whose superpolynomial part depends only on c. We also consider polynomial time preprocessing routines for EE and show that these routines cannot yield instances whose size depends polynomially only on either of the parameters b, c, k unless coNP is contained in NP/poly.