Searching for a Feedback Vertex Set with the link-cut tree

In this paper we study the Feedback Vertex Set problem of an undirected graph. This problem has several practical applications and is know to be NP-Complete. We consider this problem from its dual formulation which consists in finding a largest subset of vertexes whose induced subgraph is a forest, i.e., contains no cycles. Our main goal is to explore how the link-cut tree data structure can be used in this context. This data structure makes it possible to identify and resolve cycles quickly. This property is useful for this problem and allows us to support a simple simulated annealing algorithm. We used the PACE 2016 dataset to test the performance of the resulting algorithm. The worst approximation ratio obtained was about 1.21. The algorithm is very fast. The longest execution time was 103 seconds. To justify the importance of escaping local minimums we also present a simple greedy algorithm.