In a clustering problem one has to partition a set of elements into homogeneous and well-separated subsets. From a graph theoretic point of view, a cluster graph is a vertex-disjoint union of cliques. The clustering problem is the task of making fewest changes to the edge set of an input graph so that it becomes a cluster graph. We study the complexity of three variants of the problem. In the Cluster Completion variant edges can only be added. In Cluster Deletion, edges can only be deleted. In Cluster Editing, both edge additions and edge deletions are allowed. We also study these variants when the desired solution must contain a prespecified number of clusters.

Abstract. It is known to be NP-hard to decide whether a graph can be made chordal by the deletion of k vertices. Here we present a uniformly polynomial-time algorithm for the problem: the running time is f(k) ·n α for some constant α not depending on k and some f depending only on k. For large values of n, such an algorithm is much better than trying all the O(n k) possibilities. Therefore, the chordal deletion problem parameterized by the number k of vertices to be deleted is fixed-parameter tractable. This answers an open question of Cai [2]. 1

We present a 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.

We consider the parameterized complexity of the following problem under the flamework introduced by Downey and Fellows[4]: Given a graph G, an integer parmneter : and a non-trivial hereditary property H, are there vertices of G that induce a subgraph with property H? This problem has been proved NP-hard by Lewis and Yanna- kakis[9]. e show that if H includes all independent sets but not all cliques or vice versa, then the problem is hard for the parameterized class kV[1] and is fixed parameter tractable otherwise. In the ibrmer case, if the tbrbidden set of the property is finite, we show, in fact, that the probleln is W[1]-complete (see [] for definitions). Our prooil, both of the tractability as well as the hardness ones, involve clever use of Ramsey nmnbers.

We present e#cient fixed-parameter algorithms for the NP-complete edge modification problems Cluster Editing and Cluster Deletion.

A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following algorithmic problem, called the edge-deletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E ′ P (G). The first result of this paper states that the edge-deletion problem can be efficiently approximated for any monotone property. • For any fixed ɛ> 0 and any monotone property P, there is a deterministic algorithm, which given a graph G = (V, E) of size n, approximates E ′ P (G) in linear time O(|V | + |E|) to within an additive error of ɛn2. Given the above, a natural question is for which monotone properties one can obtain better additive approximations of E ′ P. Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist. 1. If there is a bipartite graph that does not satisfy P, then there is a δ> 0 for which it is

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For a chordal graph G = (V, E), we study the problem of whether a new vertex u � ∈ V and a given set of edges between u and vertices in V can be added to G so that the resulting graph remains chordal. We show how to resolve this efficiently, and at the same time, if the answer is no, specify a maximal subset of the proposed edges that can be added along with u, or conversely, a minimal set of extra edges that can be added in addition to the given set, so that the resulting graph is chordal. In order to do this, we give a new characterization of chordal graphs and, for each potential new edge uv, a characterization of the set of edges incident to u that also must be added to G along with uv. We propose a data structure that can compute and add each such set in O(n) time. Based on these results, we present an algorithm that computes both a minimal triangulation and a maximal chordal subgraph of an arbitrary input graph in O(nm) time, using a totally new vertex incremental approach. In contrast to previous algorithms, our process is on-line in that each new vertex is added without reconsidering any choice made at previous steps, and without requiring any knowledge of the vertices that might be added subsequently. 1

We give an algorithm with runtime O(k 2k n 3 m) for the NP-complete problem [GT35 in 6] of deciding whether a graph on n vertices and m edges can be turned into an interval graph by adding at most k edges. We thus prove that this problem is fixed parameter tractable (FPT), settling a long-standing open problem [13, 5, 19, 11]. The problem has applications in Physical Mapping of DNA [9] and in Profile Minimization for Sparse Matrix Computations [7, 20]. For the first application, our results show tractability for the case of a small number k of false negative errors, and for the second, a small number k of zero elements in the envelope.

The k-Leaf Power recognition problem is a particular case of graph power problems: For a given graph it asks whether there exists an unrooted tree—the k-leaf root—with leaves one-to-one labeled by the graph vertices and where the leaves have distance at most k iff their corresponding vertices in the graph are connected by an edge. Here we study “error correction ” versions of k-Leaf Power recognition—that is, adding or deleting at most l edges to generate a graph that has a k-leaf root. We provide several NP-completeness results in this context, and we show that the NP-complete Closest 3-Leaf Power problem (the error correction version of 3-Leaf Power) is fixed-parameter tractable with respect to the number of edge modifications or vertex deletions in the given graph. Thus, we provide the seemingly first nontrivial positive algorithmic results in the field of error compensation for leaf power problems with k> 2. To this end, as a result of independent interest, we develop a forbidden subgraph characterization of graphs with 3-leaf roots.