Increasing arc-connectivity by bounded- and fixed-size inversions

2026-04-24

AI Summary

This study investigates how to achieve $k$-arc-strong connectivity in directed graphs by reversing all arcs within vertex subsets of either fixed size or bounded size. It provides the first systematic analysis of the capability of these two types of local reversal operations to enhance arc connectivity. For fixed-size reversals, the paper establishes a structural characterization that guarantees $k$-arc-strong connectivity in sufficiently large digraphs. For bounded-size reversals, it presents a $(4k - 2 + \varepsilon)$-approximation algorithm and proves that the problem is NP-hard, APX-hard, and W[1]-hard. Bridging graph theory, combinatorial optimization, and parameterized complexity, this work delineates the theoretical limits and algorithmic potential of manipulating local structures to strengthen global connectivity.

Technology Category

• Search and Optimization: Local Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

For a digraph $D$ and some $X \subseteq V(D)$, the inversion of $X$ is the operation of flipping all arcs both of whose endvertices are in $X$. We initiate the study of establishing arc-connectivity properties by applying inversions of bounded or fixed size. For fixed-size inversions, the feasibility problem is interesting. For all integers $p \geq 2$ and $k \geq 1$, we give a characterization of the digraphs that can be made $k$-arc-strong by applying inversions of size exactly $p$, provided they are sufficiently large. For bounded-size inversions, the feasibility problem is easy, so we focus on minimising the number of inversions. We prove that for all integers $p\geq 3$ and $k \geq 1$ and any $ε>0$, there exists a polynomial-time $(4k-2+ε)$-approximation algorithm for computing the minimum number of inversions of size at most $p$ that make a given digraph $k$-arc-strong. This is in stark contrast to other results on inversion optimization problems. On the other hand, we show that for any $p\geq 3$ and $k \geq 1$ the problem is NP-hard, and, moreover, APX-hard. As a result on parameterized complexity, we show that for any $k \geq 2$, it is $W[1]$-hard with respect to $p$ to decide whether a given digraph can be made $k$-arc-strong by applying a single inversion of size at most $p$. We also prove that for a given multidigraph, it is $W[1]$-hard with respect to $\ell$ to decide whether it can be made 2-arc-strong by applying $\ell$ inversions of size 2.

Problem

• arc-connectivity
• inversion
• digraph
• k-arc-strong
• bounded-size

Innovation

• arc-connectivity
• inversion operation
• approximation algorithm
• parameterized complexity
• NP-hardness