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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)

A central problem in parameterized algorithms is to obtain algorithms with running time <i>f</i>(<i>k</i>) · <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)

- Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk +2 others
- 2015

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 - ∈)<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)

We introduce a generic algorithmic technique and apply it on decision and counting versions of graph coloring. Our approach is based on the following idea: either a graph has nice (from the algorithmic point of view) properties which allow a simple recursive procedure to find the solution fast, or the pathwidth of the graph is small, which in turn can be… (More)

In parameterized complexity each problem instance comes with a parameter k, and a parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance to an instance with size polynomial in k. Many problems have been shown to admit polynomial kernels, but it is only recently that a… (More)

The <i>k</i>-Leaf Out-Branching problem is to find an out-branching, that is a rooted oriented spanning tree, with at least <i>k</i> leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the <i>k</i>-Leaf-Out-Branching problem. We give the… (More)