Learn More
A central problem in parameterized algorithms is to obtain algorithms with running time <i>f</i>(<i>k</i>) &#183; <i>n</i><sup><i>O</i>(1)</sup> such that <i>f</i> is as slow growing function of the parameter <i>k</i> as possible. In particular, the first natural goal is to make <i>f</i>(<i>k</i>) single-exponential, that is, <i>c</i><sup><i>k</i></sup> for(More)
Branch & Reduce and dynamic programming on graphs of bounded treewidth are among the most common and powerful techniques used in the design of moderately exponential time exact algorithms for NP hard problems. In this paper we discuss the efficiency of simple algorithms based on combinations of these techniques. The idea behind these algorithms is very(More)
We investigate the parameterized complexity of <scp>Vertex Cover</scp> parameterized by the difference between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combining previously known preprocessing rules with the(More)
We obtain a number of lower bounds on the running time of algorithms solving problems on graphs of bounded treewidth. We prove the results under the Strong Exponential Time Hypothesis of Impagliazzo and Paturi. In particular, assuming that SAT cannot be solved in (2 - &#8712;)<sup><i>n</i></sup><i>m</i><sup><i>O</i>(1)</sup> time, we show that for any(More)
Let M = (E, I) be a matroid and let S = {S 1 ,. .. , S t } be a family of subsets of E of size p. A subfamily S ⊆ S is q-representative for S if for every set Y ⊆ E of size at most q, if there is a set X ∈ S disjoint from Y with X ∪ Y ∈ I, then there is a set X ∈ S disjoint from Y with X ∪ Y ∈ I. By the classical result of Bollobás, in a uniform matroid,(More)
In a parameterized problem, every instance <i>I</i> comes with a positive integer <i>k</i>. The problem is said to admit a polynomial kernel if, in polynomial time, one can reduce the size of the instance <i>I</i> to a polynomial in <i>k</i> while preserving the answer. In this work, we give two meta-theorems on kernelization. The first theorem says that(More)
It is known that the problem of deleting at most k vertices to obtain a proper interval graph (Proper Interval Vertex Deletion) is fixed parameter tractable. However, whether the problem admits a polynomial kernel or not was open. Here, we answers this question in affirmative by obtaining a polynomial kernel for Proper Interval Vertex Deletion. This(More)
In the Subset Feedback Vertex Set (Subset FVS) problem, the input is a graph G on n vertices and m edges, a subset of vertices T , referred to as terminals, and an integer k. The objective is to determine whether there exists a set of at most k vertices intersecting every cycle that contains a terminal. The study of parameterized algorithms for this(More)