Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

Given a graph $G$ and an integer $k$, the Feedback Vertex Set (FVS) problem asks if there is a vertex set $T$ of size at most $k$ that hits all cycles in the graph. The fixed-parameter tractability status of FVS in directed graphs was a long-standing open problem until Chen et al. (STOC '08) showed that it is FPT by giving a $4^{k}k!n^{O(1)}$ time algorithm. In the subset versions of this problems, we are given an additional subset $S$ of vertices (resp., edges) and we want to hit all cycles passing through a vertex of $S$ (resp. an edge of $S$). Recently, the Subset Feedback Vertex Set in undirected graphs was shown to be FPT by Cygan et al. (ICALP '11) and independently by Kakimura et al. (SODA '12). We generalize the result of Chen et al. (STOC '08) by showing that Subset Feedback Vertex Set in directed graphs can be solved in time $2^{O(k^3)}n^{O(1)}$. By our result, we complete the picture for feedback vertex set problems and their subset versions in undirected and directed graphs. Besides proving the fixed-parameter tractability of Directed Subset Feedback Vertex Set, we reformulate the random sampling of important separators technique in an abstract way that can be used for a general family of transversal problems. Moreover, we modify the probability distribution used in the technique to achieve better running time; in particular, this gives an improvement from $2^{2^{O(k)}}$ to $2^{O(k^2)}$ in the parameter dependence of the Directed Multiway Cut algorithm of Chitnis et al. (SODA '12).Comment: To appear in ACM Transactions on Algorithms. A preliminary version appeared in ICALP '12. We would like to thank Marcin Pilipczuk for pointing out a missing case in the conference version which has been considered in this version. Also, we give an single exponential FPT algorithm improving on the double exponential algorithm from the conference versio

Parameterized Algorithms for Generalizations of Directed Feedback Vertex Set

The Directed Feedback Vertex Set (DFVS) problem takes as input a directed graph~$G$ and seeks a smallest vertex set~$S$ that hits all cycles in $G$. This is one of Karp's 21 $\mathsf{NP}$-complete problems. Resolving the parameterized complexity status of DFVS was a long-standing open problem until Chen et al. [STOC 2008, J. ACM 2008] showed its fixed-parameter tractability via a $4^kk! n^{\mathcal{O}(1)}$-time algorithm, where $k = |S|$. Here we show fixed-parameter tractability of two generalizations of DFVS: - Find a smallest vertex set $S$ such that every strong component of $G - S$ has size at most~$s$: we give an algorithm solving this problem in time $4^k(ks+k+s)!\cdot n^{\mathcal{O}(1)}$. This generalizes an algorithm by Xiao [JCSS 2017] for the undirected version of the problem. - Find a smallest vertex set $S$ such that every non-trivial strong component of $G - S$ is 1-out-regular: we give an algorithm solving this problem in time $2^{\mathcal{O}(k^3)}\cdot n^{\mathcal{O}(1)}$. We also solve the corresponding arc versions of these problems by fixed-parameter algorithms

Kernels for Feedback Arc Set In Tournaments

A tournament T=(V,A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T' on O(k) vertices. In fact, given any fixed e>0, the kernelized instance has at most (2+e)k vertices. Our result improves the previous known bound of O(k^2) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k-FAST

Directed Graphs: Fixed-Parameter Tractability & Beyond

Most interesting optimization problems on graphs are NP-hard, implying that (unless P=NP) there is no polynomial time algorithm that solves all the instances of an NP-hard problem exactly. However, classical complexity measures the running time as a function of only the overall input size. The paradigm of parameterized complexity was introduced by Downey and Fellows to allow for a more refined multivariate analysis of the running time. In parameterized complexity, each problem comes along with a secondary measure k which is called the parameter. The goal of parameterized complexity is to design efficient algorithms for NP-hard problems when the parameter k is small, even if the input size is large. Formally, we say that a parameterized problem is fixed-parameter tractable (FPT) if instances of size n and parameter k can be solved in f(k).nO(1) time, where f is a computable function which does not depend on n. A parameterized problem belongs to the class XP if instances of size n and parameter k can be solved in f(k).nO(g(k)) time, where f and g are both computable functions. In this thesis we focus on the parameterized complexity of transversal and connectivity problems on directed graphs. This research direction has been hitherto relatively unexplored: usually the directed version of the problems require significantly different and more involved ideas than the ones for the undirected version. Furthermore, for directed graphs there are no known algorithmic meta-techniques: for example, there is no known algorithmic analogue of the Graph Minor Theory of Robertson and Seymour for directed graphs. As a result, the fixed-parameter tractability status of the directed versions of several fundamental problems such as Multiway Cut, Multicut, Subset Feedback Vertex Set, Odd Cycle Transversal, etc. was open. In the first part of the thesis, we develop the framework of shadowless solutions for a general class of transversal problems in directed graphs. For this class of problems, we reduce the problem of finding a solution in FPT time to that of finding a shadowless solution. Since shadowless solutions have a good (problem-specific) structure, this provides an important first step in the design of FPT algorithms for problems on directed graphs. By understanding the structure of shadowless solutions, we are able to design the first FPT algorithms for the Directed Multiway Cut problem and the Directed Subset Feedback Vertex Set problem. In the second part of the thesis, we present tight bounds on the parameterized complexity of well-studied directed connectivity problems such as Strongly Connected Steiner Subgraph and Directed Steiner Forest when parameterized by the number of terminals/terminal pairs. We design new optimal XP algorithms for the aforementioned problems, and also prove matching lower bounds for existing XP algorithms. Most of our hardness results hold even if the underlying undirected graph is planar. Finally, we conclude with some open problems regarding the parameterized complexity of transversal and connectivity problems on directed graphs

Parameterized Directed kk-Chinese Postman Problem and kk Arc-Disjoint Cycles Problem on Euler Digraphs

In the Directed $k$-Chinese Postman Problem ($k$-DCPP), we are given a connected weighted digraph $G$ and asked to find $k$ non-empty closed directed walks covering all arcs of $G$ such that the total weight of the walks is minimum. Gutin, Muciaccia and Yeo (Theor. Comput. Sci. 513 (2013) 124--128) asked for the parameterized complexity of $k$-DCPP when $k$ is the parameter. We prove that the $k$-DCPP is fixed-parameter tractable. We also consider a related problem of finding $k$ arc-disjoint directed cycles in an Euler digraph, parameterized by $k$. Slivkins (ESA 2003) showed that this problem is W[1]-hard for general digraphs. Generalizing another result by Slivkins, we prove that the problem is fixed-parameter tractable for Euler digraphs. The corresponding problem on vertex-disjoint cycles in Euler digraphs remains W[1]-hard even for Euler digraphs