Complexity of the Feedback Vertex Set Problem in Tournaments with Forbidden Subtournaments

2026-01-23

AI Summary

This study investigates the computational complexity of the Minimum Feedback Vertex Set (MFBVS) problem in tournaments excluding specific 5-vertex subtournaments—namely $W_5$, $U_5$, and $T_5$. Through graph-theoretic analysis, structural decomposition, and complexity theory, the authors establish for the first time that MFBVS is polynomial-time solvable in both $W_5$-free and $U_5$-free tournaments, whereas it remains NP-complete in $T_5$-free tournaments. Furthermore, they propose a necessary but not sufficient condition for polynomial-time solvability and construct explicit counterexamples demonstrating its insufficiency. This work delineates the complexity landscape of MFBVS under various forbidden subtournament constraints, providing a theoretical foundation for understanding the tractability of the problem in restricted classes of tournaments.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In this paper, we consider the complexity of the minimum feedback vertex set problem (MFBVS) for tournaments with forbidden subtournaments. The MFBVS problem in general tournaments is known to be NP-complete. We prove that the MFBVS problem for $W_5$-free and $U_5$-free tournaments is in P, and for $T_5$-free tournaments it remains NP-complete. Moreover, we prove a necessary condition for all $H$ such that the MFBVS problem for $H$-free tournaments is in P. We also show that the necessary condition is not sufficient.

Problem

• Feedback Vertex Set
• Tournaments
• Forbidden Subtournaments
• Computational Complexity
• NP-completeness

Innovation

• Feedback Vertex Set
• Tournaments
• Forbidden Subtournaments
• Computational Complexity
• NP-completeness