On the Parameterized Complexity of Eulerian Strong Component Arc Deletion

In this paper, we study the Eulerian Strong Component Arc Deletion problem, where the input is a directed multigraph and the goal is to delete the minimum number of arcs to ensure every strongly connected component of the resulting digraph is Eulerian. This problem is a natural extension of the Dire...

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Bibliographic Details
Published in:Algorithmica Vol. 87; no. 11; pp. 1669 - 1709
Main Authors: Blažej, Václav, Jana, Satyabrata, Ramanujan, M. S., Strulo, Peter
Format: Journal Article
Language:English
Published: New York Springer US 01.11.2025
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Complexity
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Deletion
Graph theory
Lower bounds
Mathematics of Computing
Parameterization
Parameters
Theory of Computation
Eulerian graphs
Parameterized complexity
Treewidth
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:In this paper, we study the Eulerian Strong Component Arc Deletion problem, where the input is a directed multigraph and the goal is to delete the minimum number of arcs to ensure every strongly connected component of the resulting digraph is Eulerian. This problem is a natural extension of the Directed Feedback Arc Set problem and is also known to be motivated by certain scenarios arising in the study of housing markets. The complexity of the problem, when parameterized by solution size (i.e., size of the deletion set), has remained unresolved and has been highlighted in several papers. In this work, we answer this question by ruling out (subject to the usual complexity assumptions) a fixed-parameter algorithm (FPT algorithm) for this parameter and conduct a broad analysis of the problem with respect to other natural parameterizations. We prove both positive and negative results. Among these, we demonstrate that the problem is also hard (W[1]-hard or even para-NP-hard) when parameterized by either treewidth or maximum degree alone. Complementing our lower bounds, we establish that the problem is in XP when parameterized by treewidth and FPT when parameterized either by both treewidth and maximum degree or by both treewidth and solution size. We show that on simple digraphs, these algorithms have near-optimal asymptotic dependence on the treewidth assuming the Exponential Time Hypothesis.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-025-01336-6