Making an oriented graph acyclic using inversions of bounded or prescribed size

2025-11-27

AI Summary

This study investigates the optimization and decision problem of making a directed graph acyclic via *vertex flipping*—i.e., reversing all outgoing edges of a subset of vertices. For fixed-size subsets of size (p), we systematically characterize the expressive power of such operations on the feedback arc set (FAS): even (p) yields a linear upper bound on FAS size, whereas no such bound exists for odd (p); this insight enables a polynomial-time algorithmic framework. We introduce and characterize ((=p))-reversibility—the property that exactly (p) vertices suffice to eliminate all cycles—and provide a polynomial-time decision algorithm for (p eq n-1). We prove the general problem is NP-hard and W[1]-hard. For tournament graphs, we design a kernelization algorithm yielding a polynomial kernel in the parameter (p + k) (where (k) is the minimum number of flips required). Our results unify the complexity-theoretic understanding and algorithmic tractability boundaries of the vertex-flip model.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Given an oriented graph $D$, the inversion of a subset $X$ of vertices consists in reversing the orientation of all arcs with both endpoints in $X$. When the subset $X$ is of size $p$ (resp. at most $p$), this operation is called an $(=p)$-inversion (resp. $(leq p)$-inversion). Then, an oriented graph is $(=p)$-invertible if it can be made acyclic by a sequence of $p$-inversions. We observe that, for $n=|V(D)|$, deciding whether $D$ is $(=n-1)$-invertible is equivalent to deciding whether $D$ is acyclically pushable, and thus NP-complete. In all other cases, when $p eq n-1$, we construct a polynomial-time algorithm to decide $(=p)$-invertibility. We then consider the $(= p)$-inversion number, $ ext{inv}^{= p}(D)$ (resp. $(leq p)$-inversion number, $ ext{inv}^{leq p}(D)$), defined as the minimum number of $(=p)$-inversions (resp. $(leq p)$-inversions) rendering $D$ acyclic. We show that every $(=p)$-invertible digraph $D$ satisfies $ ext{inv}^{= p}(D) leq |A(D)|$ for every integer $pgeq 2$. When $p$ is even, we bound $ ext{inv}^{= p}$ by a (linear) function of the feedback arc set number, and rule out the existence of any bounding function for odd $p$. Finally, we study the complexity of deciding whether the $(= p)$-inversion number, or the $(leq p)$-inversion number, of a given oriented graph is at most a given integer $k$. For any fixed positive integer $p geq 2$, when $k$ is part of the input, we show that both problems are NP-hard even in tournaments. In general oriented graphs, we prove $W[1]$-hardness for both problems when parameterized by $p$, even for $k=1$. In contrast, we exhibit polynomial kernels in $p + k$ for both problems in tournaments.

Problem

• Decides if oriented graphs become acyclic via bounded-size vertex subset inversions.
• Computes minimum inversion count to make oriented graphs acyclic efficiently.
• Analyzes complexity of inversion number problems in tournaments and general graphs.

Innovation

• Polynomial-time algorithm for p-invertibility when p ≠ n-1
• Bounding inversion number by feedback arc set for even p
• NP-hardness for inversion number decision in tournaments