Given an undirected graph G, a collection {(s 1 , t 1 ), . . . , (s k , t k )} of pairs of vertices, and an integer p, the Edge Multicut problem asks if there is a set S of at most p edges such that the removal of S disconnects every s i from the corresponding t i . Vertex Multicut is the analogous problem where S is a set of at most p vertices. Our main result is that both problems can be solved in time 2 O(p 3 ) · n O(1) , i.e., fixed-parameter tractable parameterized by the size p of the cutset in the solution. By contrast, it is unlikely that an algorithm with running time of the form f (p) · n O(1) exists for the directed version of the problem, as we show it to be W[1]-hard parameterized by the size of the cutset. Introduction. From the classical results of Ford and Fulkerson on minimum s − t cuts [20] to the more recent O(√ log n)-approximation algorithms for sparsest cut problems [44,1,18], the study of cut and separation problems has a deep and rich theory. One well-studied problem in this area is the Edge Multicut problem: given a graph G and pairs of vertices (s 1 , t 1 ), . . . , (s k , t k ), remove a minimum set of edges such that every s i is disconnected from its corresponding t i for every 1 ≤ i ≤ k. For k = 1, Edge Multicut is the classical s − t cut problem and can be solved in polynomial time. For k = 2, Edge Multicut remains polynomial-time solvable [46], but it becomes NP-hard for every fixed k ≥ 3 [15]. Edge Multicut can be approximated within a factor of O(log k) in polynomial time [22] (even in the weighted case, where the goal is to minimize the total weight of the removed edges). However, under the unique games conjecture of Khot [29], no constant factor approximation is possible [7]. One can analogously define the Vertex Multicut problem, where the task is to remove a minimum set of vertices. An easy reduction shows that the vertex version is more general than the edge version.Using brute force, one can decide in time n O(p) if a solution of size at most p exists. Our main result is a more efficient exact algorithm for small values of p (the O * notation hides factors that are polynomial in the input size).

The (parameterized) FEEDBACK VERTEX SET problem on directed graphs (i.e., the DFVS problem) is defined as follows: given a directed graph G and a parameter k, either construct a feedback vertex set of at most k vertices in G or report that no such a set exists. It has been a well-known open problem in parameterized computation and complexity whether the DFVS problem is fixed-parameter tractable, that is, whether the problem can be solved in time f (k)n O(1) for some function f . In this article, we develop new algorithmic techniques that result in an algorithm with running time 4 k k!n O(1) for the DFVS problem. Therefore, we resolve this open problem.

We present a method for reducing the treewidth of a graph while preserving all of its minimal s − t separators up to a certain fixed size k. This technique allows us to solve s − t Cut and Multicut problems with various additional restrictions (e.g., the vertices being removed from the graph form an independent set or induce a connected graph) in linear time for every fixed number k of removed vertices.Our results have applications for problems that are not directly defined by separators, but the known solution methods depend on some variant of separation. For example, we can solve similarly restricted generalizations of Bipartization (delete at most k vertices from G to make it bipartite) in almost linear time for every fixed number k of removed vertices. These results answer a number of open questions in the area of parameterized complexity. Furthermore, our technique turns out to be relevant for (H,C, K)and (H,C, ≤K)-coloring problems as well, which are cardinality constrained variants of the classical H-coloring problem. We make progress in the classification of the parameterized complexity of these problems by identifying new cases that can be solved in almost linear time for every fixed cardinality bound. * A subset of the results was presented at STACS 2010 [52].

The (parameterized) feedback vertex set problem on directed graphs, which we refer to as the dfvs problem, is defined as follows: given a directed graph G and a parameter k, either construct a feedback vertex set of at most k vertices in G or report that no such set exists. Whether or not the dfvs problem is fixed-parameter tractable has been a well-known open problem in parameterized computation and complexity, i.e., whether the problem can be solved in time f (k)n O(1) for some function f . In this paper we develop new algorithmic techniques that result in an algorithm with running time 4 k k!n O(1) for the dfvs problem, thus showing that this problem is fixed-parameter tractable.

We consider the following problem. Given a 2-cnf formula, is it possible to remove at most k clauses so that the resulting 2-cnf formula is satisfiable? This problem is known to different research communities in theoretical computer science under the names Almost 2-SAT, All-but-k 2-SAT, 2-cnf deletion, and 2-SAT deletion. The status of the fixed-parameter tractability of this problem is a long-standing open question in the area of parameterized complexity. We resolve this open question by proposing an algorithm that solves this problem in O(15 k × k × m 3 ) time showing that this problem is fixed-parameter tractable.