Showing papers by "Yngve Villanger published in 2011"

Subexponential Parameterized Algorithm for Minimum Fill-in

Abstract: The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. Kaplan, Shamir, and Tarjan [FOCS 1994] have shown that the problem is solvable in time O(2^(O(k)) + k2 * nm) on graphs with n vertices and m edges and thus is fixed parameter tractable. Here, we give the first subexponential parameterized algorithm solving Minimum Fill-in in time O(2^(O(\sqrt{k} log k)) + k2 * nm). This substantially lower the complexity of the problem. Techniques developed for Minimum Fill-in can be used to obtain subexponential parameterized algorithms for several related problems including Minimum Chain Completion, Chordal Graph Sandwich, and Triangulating Colored Graph.

Parameterized complexity of vertex deletion into perfect graph classes

Abstract: Vertex deletion problems are at the heart of parameterized complexity. For a graph class F, the F-Deletion problem takes as input a graph G and an integer k. The question is whether it is possible to delete at most k vertices from G such that the resulting graph belongs to F. Whether Perfect Deletion is fixed-parameter tractable, and whether Chordal Deletion admits a polynomial kernel, when parameterized by k, have been stated as open questions in previous work. We show that Perfect Deletion (k) and Weakly Chordal Deletion (k) are W[2]-hard. In search of positive results, we study restricted variants such that the deleted vertices must be taken from a specified set X, which we parameterize by |X|. We show that for Perfect Deletion and Weakly Chordal Deletion, although this restriction immediately ensures fixed parameter tractability, it is not enough to yield polynomial kernels, unless NP ⊆ coNP/poly. On the positive side, for Chordal Deletion, the restriction enables us to obtain a kernel with O(|X|4) vertices.

Faster Parameterized Algorithms for Minimum Fill-in

Abstract: We present two parameterized algorithms for the Minimum Fill-in problem, also known as Chordal Completion: given an arbitrary graph G and integer k, can we add at most k edges to G to obtain a chordal graph? Our first algorithm has running time $\mathcal {O}(k^{2}nm+3.0793^{k})$, and requires polynomial space. This improves the base of the exponential part of the best known parameterized algorithm time for this problem so far. We are able to improve this running time even further, at the cost of more space. Our second algorithm has running time $\mathcal {O}(k^{2}nm+2.35965^{k})$ and requires $\mathcal {O}^{\ast}(1.7549^{k})$ space. To achieve these results, we present a new lemma describing the edges that can safely be added to achieve a chordal completion with the minimum number of edges, regardless of k.

Exact algorithm for the maximum induced planar subgraph problem

Abstract: We prove that in an n-vertex graph, an induced planar subgraph of maximum size can be found in time O(1.7347n). This is the first algorithm breaking the trivial 2nnO(1) bound of the brute-force search algorithm for the MAXIMUM INDUCED PLANAR SUBGRAPH problem.

Enumerating minimal subset feedback vertex sets

Abstract: The SUBSET FEEDBACK VERTEX SET problem takes as input a weighted graph G and a vertex subset S of G, and the task is to find a set of vertices of total minimum weight to be removed from G such that in the remaining graph no cycle contains a vertex of S. This problem is a generalization of two classical NP-complete problems: FEEDBACK VERTEX SET and MULTIWAY CUT. We show that it can be solved in time O(1.8638n) for input graphs on n vertices. To the best of our knowledge, no exact algorithm breaking the trivial 2nnO(1)-time barrier has been known for SUBSET FEEDBACK VERTEX SET, even in the case of unweighted graphs. The mentioned running time is a consequence of the more general main result of this paper: we show that all minimal subset feedback vertex sets of a graph can be enumerated in O(1.8638n) time.

Subexponential fixed-parameter tractability of cluster editing

Abstract: In the Correlation Clustering, also known as Cluster Editing, we are given an undirected n-vertex graph G and a positive integer k. The task is to decide if G can be transformed into a cluster graph, i.e., a disjoint union of cliques, by changing at most k adjacencies, i.e. by adding/deleting at most k edges. We give a subexponential algorithm that, in time 2^O(sqrt(pk)) + n^O(1) decides whether G can be transformed into a cluster graph with p cliques by changing at most k adjacencies. We complement our algorithmic findings by the following tight lower bounds on the asymptotic behaviour of our algorithm. We show that, unless ETH fails, for any constant 0 < s <= 1, there is p = Theta(k^s) such that there is no algorithm deciding in time 2^o(sqrt(pk)) n^O(1) whether G can be transformed into a cluster graph with p cliques by changing at most k adjacencies.

Minimum Fill-in of Sparse Graphs: Kernelization and Approximation

Abstract: The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. The problem has important applications in numerical algebra, in particular in sparse matrix computations. We develop kernelization algorithms for the problem on several classes of sparse graphs. We obtain linear kernels on planar graphs, and kernels of size O(k^{3/2}) in graphs excluding some fixed graph as a minor and in graphs of bounded degeneracy. As a byproduct of our results, we obtain approximation algorithms with approximation ratios O(log{k}) on planar graphs and O(sqrt{k} log{k}) on H-minor-free graphs. These results significantly improve the previously known kernelization and approximation results for Minimum Fill-in on sparse graphs.

k-Gap Interval Graphs

Abstract: We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersection with an interval associated to the other vertex. A graph on n vertices is a k-gap interval graph if it has a multiple interval representation with at most n+k intervals in total. In order to scale up the nice algorithmic properties of interval graphs (where k=0), we parameterize graph problems by k, and find FPT algorithms for several problems, including Feedback Vertex Set, Dominating Set, Independent Set, Clique, Clique Cover, and Multiple Interval Transversal. The Coloring problem turns out to be W[1]-hard and we design an XP algorithm for the recognition problem.