Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Sophie Spirkl, Yun Xing

TL;DR

The paper tackles the minimum feedback vertex set problem in tournaments with forbidden subtournaments, focusing on a five-vertex forbidden family. It develops a reduction framework to prime tournaments and leverages structural characterizations for $W_5$-free and $U_5$-free cases to achieve polynomial-time algorithms, while proving NP-completeness for $T_5$-free tournaments. A key contribution is a necessary condition for when the problem can be in $P$ for $H$-free tournaments, and the demonstration that this condition is not sufficient. The work combines backedge-graph-based reductions, prime-tournament structure theorems, and dynamic programming over tri-partitions to delineate the boundary between tractability and hardness in this natural directed-graph setting with forbidden subtournaments.

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.

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

TL;DR

The paper tackles the minimum feedback vertex set problem in tournaments with forbidden subtournaments, focusing on a five-vertex forbidden family. It develops a reduction framework to prime tournaments and leverages structural characterizations for -free and -free cases to achieve polynomial-time algorithms, while proving NP-completeness for -free tournaments. A key contribution is a necessary condition for when the problem can be in for -free tournaments, and the demonstration that this condition is not sufficient. The work combines backedge-graph-based reductions, prime-tournament structure theorems, and dynamic programming over tri-partitions to delineate the boundary between tractability and hardness in this natural directed-graph setting with forbidden subtournaments.

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 -free and -free tournaments is in P, and for -free tournaments it remains NP-complete. Moreover, we prove a necessary condition for all such that the MFBVS problem for -free tournaments is in P. We also show that the necessary condition is not sufficient.
Paper Structure (12 sections, 21 theorems, 1 equation, 2 algorithms)

This paper contains 12 sections, 21 theorems, 1 equation, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Motivation
  4. Our Main Results
  5. Polynomial-time algorithms
  6. Polynomial-time reduction to prime tournaments
  7. Excluding $W_5$
  8. Excluding $U_5$
  9. NP-completeness Results
  10. Excluding 1-degenerate Tournaments
  11. Excluding $k$-snakes
  12. Acknowledgement

Key Result

Proposition 1

Let $G$ be a tournament. The following are equivalent: (1) $G$ is transitive. (2) $G$ contains no cyclic triangle. (3) $G$ is acylic. (4) $G$ admits a topological ordering on its vertices $V(G)$. (5) There exists an ordering of $V(G)$ that gives a backedge graph with no edges.

Theorems & Definitions (28)

  • Proposition 1: Folklore
  • Definition 1.1.1: Feedback vertex set
  • Theorem 1.1: Speckenmeyer, SPECKENMEYER
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.3.1
  • Theorem 1.5
  • Corollary 1.3.2
  • Theorem 2.1.1
  • ...and 18 more