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In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph $D$ on $n$ vertices and $m$ edges, and an integer $k$. The objective is to determine whether there exists a set of at most $k$ vertices intersecting every directed cycle of $D$. Whether or not DFVS admits a fixed parameter tractable (FPT) algorithm was considered the most important open problem in parameterized complexity until Chen, Liu, Lu, OSullivan and Razgon [JACM 2008] answered the question in the affirmative. They gave an algorithm for the problem with running time $O(k!4^kk^4nm)$. Since then, no faster algorithm for the problem has been found. In this paper, we give an algorithm for DFVS with running time $O(k!4^kk^5(n+m))$. Our algorithm is the first algorithm for DFVS with linear dependence on input size. Furthermore, the asymptotic dependence of the running time of our algorithm on the parameter $k$ matches up to a factor $k$ the algorithm of Chen, Liu, Lu, OSullivan and Razgon. On the way to designing our algorithm for DFVS, we give a general methodology to shave off a factor of $n$ from iterative-compression based algorithms for a few other well-studied covering problems in parameterized complexity. We demonstrate the applicability of this technique by speeding up by a factor of $n$, the current best FPT algorithms for Multicut [STOC 2011, SICOMP 2014] and Directed Subset Feedback Vertex Set [ICALP 2012, TALG 2014].
The optimization version of the Unique Label Cover problem is at the heart of the Unique Games Conjecture which has played an important role in the proof of several tight inapproximability results. In recent years, this problem has been also studied
In this paper we consider Simultaneous Feedback Edge Set (Sim-FES) problem. In this problem, the input is an $n$-vertex graph $G$, an integer $k$ and a coloring function ${sf col}: E(G) rightarrow 2^{[alpha]}$ and the objective is to check whether th
The Cut & Count technique and the rank-based approach have lead to single-exponential FPT algorithms parameterized by treewidth, that is, running in time $2^{O(tw)}n^{O(1)}$, for Feedback Vertex Set and connect
We study the recently introduced Connected Feedback Vertex Set (CFVS) problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical Feedback Vertex Set problem and is defined as follows: given a graph G=(V,E)
A cactus is a connected graph that does not contain $K_4 - e$ as a minor. Given a graph $G = (V, E)$ and integer $k ge 0$, Cactus Vertex Deletion (also known as Diamond Hitting Set) is the problem of deciding whether $G$ has a vertex set of size at m