Dynamic Parameterized Feedback Problems in Tournaments
Anna Zych-Pawlewicz, Marek Żochowski
TL;DR
This work provides the first dynamic, parameterized algorithms for Feedback Arc Set in Tournaments (FAST) and Feedback Vertex Set in Tournaments (FVST) under arc reversals. The authors develop two triangle-detection data structures, $\mathbb{DTP}[n]$ and $\mathbb{DT}[n]$, to support a branching framework that uses triangles to guide deletions or reversals, with performance depending on the number of arc-disjoint triangles $\mathsf{ADT}(T)$. For FVST, they introduce a tokenized, heavy-vertex kernelization approach via $\mathbb{DREM}[n,k]$ and a long-arc decomposition, wrapped by $\mathbb{DREMP}[n,k]$, enabling efficient handling of vertex removals within a promise bound. The paper delivers concrete worst-case bounds: dynamic FAST in the promise model achieves $O(\sqrt{g(K)})$ updates and $O(3^K K \sqrt{K})$ queries, while the full model attains $O(\log^2 n)$ updates and $O(3^K K \log^2 n)$ queries; dynamic FVST in the promise model achieves $O(g(K)^5)$ updates and $O(3^K g(K) K^3)$ queries. Overall, these results establish the first dynamic, parameterized algorithms for FAST and FVST in tournaments and introduce a suite of new dynamic data structures with potential impact on online ranking and voting applications.
Abstract
In this paper we present the first dynamic algorithms for the problem of Feedback Arc Set in Tournaments (FAST) and the problem of Feedback Vertex Set in Tournaments (FVST). Our algorithms maintain a dynamic tournament on n vertices altered by redirecting the arcs, and answer if the tournament admits a feedback arc set (or respectively feedback vertex set) of size at most K, for some chosen parameter K. For dynamic FAST we offer two algorithms. In the promise model, where we are guaranteed, that the size of the solution does not exceed g(K) for some computable function g, we give an $O(\sqrt{g(K)})$ update and $O(3^K K \sqrt{K})$ query algorithm. In the general setting without any promise, we offer an $O(\log^2 n)$ update and $O(3^K K \log^2 n)$ query time algorithm for dynamic FAST. For dynamic FVST we offer an algorithm working in the promise model, which admits $O(g^5(K))$ update and $O(3^K K^3 g(K))$ query time.