An FPT algorithm for cycle rank on semi-complete digraphs

2026-06-28

AI Summary

This study addresses the fixed-parameter tractability of determining whether the cycle rank of a directed graph is at most a given parameter \( w \). The authors present the first fixed-parameter tractable (FPT) algorithms for this problem on semicomplete digraphs and digraphs of bounded directed clique-width. Their approach employs dynamic programming over directed clique-width expressions, yielding a decision algorithm that runs in \( O(9^{(w+1)4^{w+2}} \cdot n^2) \) time on semicomplete digraphs and in \( O(9^{(w+1)4^{k}} \cdot n) \) time on digraphs of directed clique-width at most \( k \). Furthermore, by leveraging the cycle rank parameter, they solve the Minimum Feedback Arc Set problem in \( n^{O(w)} \) time.

Abstract

Cycle rank is a depth parameter for digraphs introduced by Eggan in 1963. Gruber (DMTCS 2012) and Giannopoulou, Hunter, and Thilikos (DAM 2012) asked whether the problem of determining if a given digraph has cycle rank at most $w$ is fixed-parameter tractable parameterized by $w$. We provide such algorithms for semi-complete digraphs, and for digraphs of bounded directed clique-width. Specifically, we show that given an $n$-vertex semi-complete digraph $G$ and an integer $w$, one can in time $\mathcal{O}(9^{(w+1)4^{w+2}} \cdot n^2)$ determine whether $G$ has cycle rank at most $w$. The proof is reduced to the case of bounded directed clique-width, and we then show that given an $n$-vertex digraph $G$ with a directed clique-width $k$-expression and an integer $w$, one can in time $\mathcal{O}(9^{(w+1) 4^k} \cdot n)$ determine whether $G$ has cycle rank at most $w$. Additionally, we consider the \textsc{Minimum Feedback Arc Set} problem on semi-complete digraphs, and show that it can be solved in time $n^{\mathcal{O}(w)}$, where $w$ is the cycle rank of the given semi-complete digraph.

Problem

• cycle rank
• semi-complete digraphs
• fixed-parameter tractability
• directed clique-width
• feedback arc set

Innovation

• cycle rank
• fixed-parameter tractable
• semi-complete digraphs
• directed clique-width
• feedback arc set