On the Parameterized Complexity of Eulerian Strong Component Arc Deletion

2024-08-25

International Symposium on Parameterized and Exact Computation

AI Summary

This paper studies the Eulerian Strongly Connected Component Arc Deletion problem on directed multigraphs: delete a minimum number of arcs so that each strongly connected component becomes Eulerian. We first prove the problem is para-NP-hard and W[1]-hard with respect to solution size $k$, assuming the Exponential Time Hypothesis (ETH), thus ruling out FPT algorithms parameterized by $k$ alone. We systematically delineate its multi-parameter complexity landscape: the problem is XP parameterized by treewidth ($mathrm{tw}$) alone, but becomes fixed-parameter tractable (FPT) when parameterized by $mathrm{tw} + Delta$ or $mathrm{tw} + k$, where $Delta$ denotes maximum degree. We design a tight algorithm matching the ETH lower bound, combining tree-width-based dynamic programming with refined reductions. Our main contribution is a complete parameterized complexity classification, revealing the interplay among vertex degrees, structural width (treewidth), and solution size in determining computational tractability.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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 tractable (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 these algorithms have near-optimal asymptotic dependence on the treewidth assuming the Exponential Time Hypothesis.

Problem

• Determine complexity of Eulerian Strong Component Arc Deletion problem
• Analyze parameterized complexity with respect to treewidth and degree
• Establish FPT and XP results for specific parameter combinations

Innovation

• Parameterized complexity analysis of Eulerian arc deletion
• FPT and XP algorithms for treewidth parameterization
• Hardness results for treewidth and degree parameters