Rolf Niedermeier

Invitation to data reduction and problem kernelization

To solve NP-hard problems, polynomial-time preprocessing is a natural and promising approach. Preprocessing is based on data reduction techniques that take a problems input instance and try to perform a reduction to a smaller, equivalent problem kernel. Problem kernelization is a methodology that is rooted in parameterized computational complexity. In this brief survey, we present data reduction and problem kernelization as a promising research field for algorithm and complexity theory.

Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs

Abstract. We present an algorithm that constructively produces a solution to the k -DOMINATING SET problem for planar graphs in time O(c^ \sqrt k n) , where c=4^ 6\sqrt 34 . To obtain this result, we show that the treewidth of a planar graph with domination number γ (G) is O(\sqrt \rule 0pt 4pt \smash γ (G) ) , and that such a tree decomposition can be found in O(\sqrt \rule 0pt 4pt \smash γ (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(c1^ \sqrt k n) time, where c1=3^ 36\sqrt 34 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.

Polynomial-time data reduction for dominating set

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.

Upper bounds for vertex cover further improved

The problem instance of Vertex Cover consists of an undirected graph G = (V, E) and a positive integer k, the question is whether there exists a subset C ⊆ V of vertices such that each edge in E has at least one of its endpoints in C with |C| ≤ k. We improve two recent worst case upper bounds for Vertex Cover. First, Balasubramanian et al. showed that Vertex Cover can be solved in time O(kn+1:32472kk2), where n is the number of vertices in G. Afterwards, Downey et al. improved this to O(kn+1:31951kk2). Bringing the exponential base significantly below 1:3, we present the new upper bound O(kn+1:29175kk2).

A general method to speed up fixed-parameter-tractable algorithms

A fixed-parameter-tractable algorithm, or FPT algorithm for short, gets an instance (I,k) as its input and has to decide whether (I,k)∈L for some parameterized problem L. Many parameterized algorithms work in two stages: reduction to a problem kernel and bounded search tree. Their time complexity is then of the form O(p(|I|)+q(k)ξk), where q(k) is the size of the problem kernel. We show how to modify these algorithms to obtain time complexity O(p(|I|)+ξk), if q(k) is polynomial.

Graph-Modeled Data Clustering: Exact Algorithms for Clique Generation

We present efficient fixed-parameter algorithms for the NP-complete edge modification problems Cluster Editing and Cluster Deletion. Here, the goal is to make the fewest changes to the edge set of an input graph such that the new graph is a vertex-disjoint union of cliques. Allowing up to k edge additions and deletions (Cluster Editing), we solve this problem in O(2.27k + |V|3) time; allowing only up to k edge deletions (Cluster Deletion), we solve this problem in O(1.77k + |V|3) time. The key ingredients of our algorithms are two easy to implement bounded search tree algorithms and an efficient polynomial-time reduction to a problem kernel of size O(k3). This improves and complements previous work. Finally, we discuss further improvements on search tree sizes using computer-generated case distinctions.

Automated Generation of Search Tree Algorithms for Graph Modification Problems

We present a (seemingly first) framework for an automated generation of exact search tree algorithms for NP-hard problems. The purpose of our approach is two-fold-rapid development and improved upper bounds. Many search tree algorithms for various problems in the literature are based on complicated case distinctions. Our approach may lead to a much simpler process of developing and analyzing these algorithms. Moreover, using the sheer computing power of machines it may also lead to improved upper bounds on search tree sizes (i.e., faster exact solving algorithms) in comparison with previously developed hand-made search trees.

On efficient fixed-parameter algorithms for weighted vertex cover

We investigate the fixed-parameter complexity of WEIGHTED VERTEX COVER. Given a graph G = (V, E), a weight function ω: V → R+, and k ∈ R+, WEIGHTED VERTEX COVER (WVC for short) asks for a vertex subset C ⊆ V of total weight at most k such that every edge of G has at least one endpoint in C. WVC and its natural variants are NP-complete. We observe that, when restricting the range of ω to positive integers, the so-called INTEGER-WVC can be solved as fast as unweighted VERTEX COVER. Our main result is that if the range of ω is restricted to positive reals ≥ 1, then so-called REAL-WVC can be solved in time O(1.3954k + k|V|). By way of contrast, unless P = NP, the problem is not fixed-parameter tractable if arbitrary weights > 0 are allowed. Using dynamic programming, at the expense of exponential memory use, we can improve the running time of REALWVC to O(1.3788k + k|V|). The same technique applied to a known algorithm yields the so far fastest algorithm for unweighted VERTEX COVER, running in time O(1.2832kk + k|V|).

Fixed-Parameter Algorithms for Cluster Vertex Deletion

We initiate the first systematic study of the NP-hard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixed-parameter algorithmics. In the unweighted case, one searches for a minimum number of vertex deletions to transform a graph into a collection of disjoint cliques. The parameter is the number of vertex deletions. We present efficient fixed-parameter algorithms for CVD applying the fairly new iterative compression technique. Moreover, we study the variant of CVD where the maximum number of cliques to be generated is prespecified. Here, we exploit connections to fixed-parameter algorithms for (weighted) Vertex Cover.

Linear problem kernels for NP-hard problems on planar graphs

We develop a generic framework for deriving linear-size problem kernels for NP-hard problems on planar graphs. We demonstrate the usefulness of our framework in several concrete case studies, giving new kernelization results for CONNECTED VERTEX COVER, MINIMUM EDGE DOMINATING SET, MAXIMUM TRIANGLE PACKING, AND EFFICIENT DOMINATING Set on planar graphs. On the route to these results, we present effective, problem-specific data reduction rules that are useful in any approach attacking the computational intractability of these problems.