We present an algorithm that constructively produces a solution to the k-dominating set problem for planar graphs in time O(c . To obtain this result, we show that the treewidth of a planar graph with domination number (G) is O( (G)), and that such a tree decomposition can be found in O( (G)n) time. The same technique can be used to show that the k-face cover problem ( find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c n) time, where c 1 = 3 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k-dominating set, e.g., k-independent dominating set and k-weighted dominating set.

Graph minors theory, developed by Robertson & Seymour, provides a list of powerful theoretical results and tools. However, the wide spread opinion in Graph Algorithms community about this theory is that it is mainly of theoretical importance.

Dealing with the NP-complete Dominating Set problem on graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar graphs has a so-called problem kernel of linear size, achieved by two simple and easy to implement reduction rules. Moreover, having implemented our reduction rules, first experiments indicate the impressive practical potential of these rules. Thus, this work seems to open up a new and prospective way how to cope with one of the most important problems in graph theory and combinatorial optimization.

Abstract not found

The goal of this article is to provide a tourist guide, with an eye towards structural issues, to what I consider some of the major highlights of parameterized complexity.

Abstract. Determining whether a parameterized problem is kernelizable and has a small kernel size has recently become one of the most interesting topics of research in the area of parameterized complexity and algorithms. Theoretically, it has been proved that a parameterized problem is kernelizable if and only if it is fixed-parameter tractable. Practically, applying a data reduction algorithm to reduce an instance of a parameterized problem to an equivalent smaller instance (i.e., a kernel) has led to very efficient algorithms and now goes hand-in-hand with the design of practical algorithms for solving NP-hard problems. Well-known examples of such parameterized problems include the vertex cover problem, which is kernelizable to a kernel of size bounded by 2k, and the planar dominating set problem, which is kernelizable to a kernel of size bounded by 335k. In this paper we develop new techniques to derive upper and lower bounds on the kernel size for certain parameterized problems. In terms of our lower bound results, we show, for example, that unless P = NP, planar vertex cover does not have a problem kernel of size smaller than 4k/3, and planar independent set and planar dominating set do not have kernels of size smaller than 2k. In terms of our upper bound results, we further reduce the upper bound on the kernel size for the planar dominating set problem to 67k, improving significantly the 335k previous upper bound given by Alber, Fellows, and Niedermeier [J. ACM, 51 (2004), pp. 363–384]. This latter result is obtained by introducing a new set of reduction and coloring rules, which allows the derivation of nice combinatorial properties in the kernelized graph leading to a tighter bound on the size of the kernel. The paper also shows how this improved upper bound yields a simple and competitive algorithm for the planar dominating set problem.

Abstract not found

We demonstrate a new connection between fixed-parameter tractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of so-called “bidimensional” problems to show that essentially all such problems have both subexponential fixed-parameter algorithms and PTASs. Bidimensional problems include e.g. feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex-removal problems, dominating set, edge dominating set, r-dominating set, diameter, connected dominating set, connected edge dominating set, and connected r-dominating set. We obtain PTASs for all of these problems in planar graphs and certain generalizations; of particular interest are our results for the two well-known problems of connected dominating set and general feedback vertex set for planar graphs and their generalizations, for which PTASs were not known to exist. Our techniques generalize and in some sense unify the two main previous approaches for designing PTASs in planar graphs, namely, the Lipton-Tarjan separator approach [FOCS’77] and the Baker layerwise decomposition approach [FOCS’83]. In particular, we replace the notion of separators with a more powerful tool from the bidimensionality theory, enabling the first approach to apply to a much broader class of minimization problems than previously possible; and through the use of a structural backbone and thickening of layers we demonstrate how the second approach can be applied to problems with a “nonlocal” structure.

Graph separation is a well-known tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many well-known NP-hard (planar) graph problems. We coin the key notion of glueable select&verify graph problems and derive from that a prospective way to easily check whether a planar graph problem will allow for a xed parameter algorithm of running time c p

We present an improved dynamic programming strategy for dominating set and related problems on graphs that are given together with a tree decomposition of width k. We obtain an O(4 n) algorithm for dominating set, where n is the number of nodes of the tree decomposition.