Igor Razgon

Almost 2-SAT 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^kxkxm^3) time showing that this problem is fixed-parameter tractable.

On the Minimum Feedback Vertex Set Problem: Exact and Enumeration Algorithms

Abstract We present a time

Fixed-parameter tractability of multicut parameterized by the size of the cutset

\mathcal {O}(1.7548^{n})

Exact computation of maximum induced forest

algorithm finding a minimum feedback vertex set in an undirected graph on n vertices. We also prove that a graph on n vertices can contain at most 1.8638n minimal feedback vertex sets and that there exist graphs having 105n/10≈1.5926n minimal feedback vertex sets.

Finding small separators in linear time via treewidth reduction

Given an undirected graph

A fixed-parameter algorithm for the directed feedback vertex set problem

G

Faster computation of maximum independent set and parameterized vertex cover for graphs with maximum degree 3

, a collection {(s₁,t₁), ..., (s_l,t_l)} of pairs of vertices, and an integer p, the Edge Multicut problem ask if there is a set <i>S</i> 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(p³) ⋅ 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.

Connected coloring completion for general graphs: algorithms and complexity

We propose a backtrack algorithm that solves a generalized version of the Maximum Induced Forest problem (MIF) in time O*(1.8899n). The MIF problem is complementary to finding a minimum Feedback Vertex Set (FVS), a well-known intractable problem. Therefore the proposed algorithm can find a minimum FVS as well. To the best of our knowledge, this is the first algorithm that breaks the O*(2n) barrier for the general case of FVS. Doing the analysis, we apply a more sophisticated measure of the problem size than the number of nodes of the underlying graph

Treewidth Reduction for Constrained Separation and Bipartization Problems

We present a method for reducing the treewidth of a graph while preserving all of its minimal <i>s</i>-<i>t</i> separators up to a certain fixed size <i>k</i>. This technique allows us to solve <i>s</i>-<i>t</i> <scp>Cut</scp> and <scp>Multicut</scp> 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 <i>k</i> 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 <scp>Bipartization</scp> (delete at most <i>k</i> vertices from <i>G</i> to make it bipartite) in almost linear time for every fixed number <i>k</i> 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 (<i>H</i>, <i>C</i>, <i>K</i>)- and (<i>H</i>, <i>C</i>,≤K)-coloring problems as well, which are cardinality constrained variants of the classical <i>H</i>-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.

Fixed-Parameter Tractability of Multicut Parameterized by the Size of the Cutset

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)nO(1) for some function f. In this paper we develop new algorithmic techniques that result in an algorithm with running time 4k k! nO(1) for the dfvs problem, thus showing that this problem is fixed-parameter tractable.