Dynamic Parameterized Feedback Problems in Tournaments

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.