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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)

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)

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)

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)

- Daniel Lokshtanov, Maria Chudnovsky, Rolf Niedermeier, Sverre Storøy, Noga Alon, Oren Ben-Zwi +28 others
- 2009

i Acknowledgements First of all, I would like to thank my supervisor Pinar Heggernes for her help and guidance, both with my reasearch and with all other aspects of being a graduate student. This thesis would not have existed without her. Thomassen and Yngve Villanger. A special thanks goes to Saket Saurabh, with whom I have shared many sleepless nights and… (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)

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

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)