Showing papers by "Pinar Heggernes published in 2011"

Contracting Graphs to Paths and Trees

Abstract: Vertex deletion and edge deletion problems play a central role in Parameterized Complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. Interestingly, the study of edge contraction problems of this type from a parameterized perspective has so far been left largely unexplored. We consider two basic edge contraction problems, which we call Path-Contractibility and Tree-Contractibility. Both problems take an undirected graph $G$ and an integer $k$ as input, and the task is to determine whether we can obtain a path or an acyclic graph, respectively, by contracting at most $k$ edges of $G$. Our main contribution is an algorithm with running time $4^{k+O(\log^2 k)} + n^{O(1)}$ for Path-Contractibility and an algorithm with running time $4.88^k n^{O(1)}$ for Tree-Contractibility, based on a novel application of the color coding technique of Alon, Yuster and Zwick. Furthermore, we show that Path-Contractibility has a kernel with at most $5k+3$ vertices, while Tree-Contractibility does not have a polynomial kernel unless coNP $\subseteq$ NP/poly. We find the latter result surprising, because of the strong connection between Tree-Contractibility and Feedback Vertex Set, which is known to have a vertex kernel with size $O(k^2)$.

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.

Obtaining a Bipartite Graph by Contracting Few Edges

Abstract: We initiate the study of the Bipartite Contraction problem from the perspective of param- eterized complexity. In this problem we are given a graph G on n vertices and an integer k, and the task is to determine whether we can obtain a bipartite graph from G by a sequence of at most k edge contractions. Our main result is an $f(k)n^{O(1)}$ time algorithm for Bipartite Con- traction. Despite a strong resemblance between Bipartite Contraction and the classical Odd Cycle Transversal (OCT) problem, the methods developed to tackle OCT do not seem to be directly applicable to Bipartite Contraction. To obtain our result, we combine several techniques and concepts that are central in parameterized complexity: iterative compression, irrelevant vertex, and important separators. To the best of our knowledge, this is the first time the irrelevant vertex technique and the concept of important separators are applied in unison. Furthermore, our algorithm may serve as a comprehensible example of the usage of the irrelevant vertex technique.

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.

Finding contractions and induced minors in chordal graphs via disjoint paths

Abstract: The k-Disjoint Paths problem, which takes as input a graph G and k pairs of specified vertices (si,ti), asks whether G contains k mutually vertex-disjoint paths Pi such that Pi connects si and ti, for i=1,…,k. We study a natural variant of this problem, where the vertices of Pi must belong to a specified vertex subset Ui for i=1,…,k. In contrast to the original problem, which is polynomial-time solvable for any fixed integer k, we show that this variant is NP-complete even for k=2. On the positive side, we prove that the problem becomes polynomial-time solvable for any fixed integer k if the input graph is chordal. We use this result to show that, for any fixed graph H, the problems H-Contractibility and H-Induced Minor can be solved in polynomial time on chordal graphs. These problems are to decide whether an input graph G contains H as a contraction or as an induced minor, respectively.

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.

Contracting graphs to paths and trees

Abstract: Vertex deletion and edge deletion problems play a central role in Parameterized Complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. The study of analogous edge contraction problems has so far been left largely unexplored from a parameterized perspective. We consider two basic problems of this type: Tree Contraction and Path Contraction. These two problems take as input an undirected graph G on n vertices and an integer k, and the task is to determine whether we can obtain an acyclic graph or a path, respectively, by a sequence of at most k edge contractions in G. We present an algorithm with running time 4.98knO(1) for Tree Contraction, based on a variant of the color coding technique of Alon, Yuster and Zwick, and an algorithm with running time 2k+o(k)+nO(1) for Path Contraction. Furthermore, we show that Path Contraction has a kernel with at most 5k+3 vertices, while Tree Contraction does not have a polynomial kernel unless NP ⊆ coNP/poly. We find the latter result surprising, because of the connection between Tree Contraction and Feedback Vertex Set, which is known to have a kernel with 4k2 vertices.

Cutwidth of Split Graphs and Threshold Graphs

Abstract: We give a linear-time algorithm to compute the cutwidth of threshold graphs, thereby resolving the computational complexity of cutwidth on this graph class Threshold graphs are a well-studied subclass of interval graphs and of split graphs, both of which are unrelated subclasses of chordal graphs To complement our result, we show that cutwidth is NP-complete on split graphs, and consequently also on chordal graphs The cutwidth of interval graphs is still open, and only very few graph classes are known so far on which polynomial-time cutwidth algorithms exist Thus we contribute to define the border between graph classes on which cutwidth is polynomially solvable and on which it remains NP-complete

Computing the Clique-width of large path powers in linear time via a new characterisation of clique-width

Abstract: Clique-width is one of the most important graph parameters, as many NP-hard graph problems are solvable in linear time on graphs of bounded clique-width. Unfortunately, the computation of clique-width is among the hardest problems. In fact, we do not know of any other algorithm than brute force for the exact computation of clique-width on any large graph class of unbounded clique-width. Another difficulty about clique-width is the lack of alternative characterisations of it that might help in coping with its hardness. In this paper, we present two results. The first is a new characterisation of clique-width based on rooted binary trees, completely without the use of labelled graphs. Our second result is the exact computation of the clique-width of large path powers in polynomial time, which has been an open problem for a decade. The presented new characterisation is used to achieve this latter result. With our result, large k-path powers constitute the first non-trivial infinite class of graphs of unbounded clique-width whose clique-width can be computed exactly in polynomial time.

Strongly chordal and chordal bipartite graphs are sandwich monotone

Abstract: A graph class is sandwich monotone if, for every pair of its graphs G 1=(V,E 1) and G 2=(V,E 2) with E 1?E 2, there is an ordering e 1,?,e k of the edges in E 2?E 1 such that G=(V,E 1?{e 1,?,e i }) belongs to the class for every i between 1 and k In this paper we show that strongly chordal graphs and chordal bipartite graphs are sandwich monotone, answering an open question by Bakonyi and Bono (Czechoslov Math J 46:577---583, 1997) So far, very few classes have been proved to be sandwich monotone, and the most famous of these are chordal graphs Sandwich monotonicity of a graph class implies that minimal completions of arbitrary graphs into that class can be recognized and computed in polynomial time For minimal completions into strongly chordal or chordal bipartite graphs no polynomial-time algorithm has been known With our results such algorithms follow for both classes In addition, from our results it follows that all strongly chordal graphs and all chordal bipartite graphs with edge constraints can be listed efficiently

Edge contractions in subclasses of chordal graphs

Abstract: Modifying a given graph to obtain another graph is a wellstudied problem with applications in many fields. Given two input graphs G and H, the Contractibility problem is to decide whether H can be obtained from G by a sequence of edge contractions. This problem is known to be NP-complete already when both input graphs are trees of bounded diameter. We prove that CONTRACTIBILITY can be solved in polynomial time when G is a trivially perfect graph and H is a threshold graph, thereby giving the first classes of graphs of unbounded treewidth and unbounded degree on which the problem can be solved in polynomial time. We show that this polynomial-time result is in a sense tight, by proving that Contractibility is NP-complete when G and H are both trivially perfect graphs, and when G is a split graph and H is a threshold graph. If the graph H is fixed and only G is given as input, then the problem is called H-CONTRACTIBILITY. This problem is known to be NP-complete on general graphs already when H is a path on four vertices. We show that, for any fixed graph H, the H-CONTRACTIBILITY problem can be solved in polynomial time if the input graph G is a split graph.

A generic approach to decomposition algorithms, with an application to digraph decomposition

Abstract: A set family is a collection of sets over a universe. If a set family satisfies certain closure properties then it admits an efficient representation of its members by labeled trees. The size of the tree is proportional to the size of the universe, whereas the number of set family members can be exponential. Computing such efficient representations is an important task in algorithm design. Set families are usually not given explicitly (by listing their members) but represented implicitly. We consider the problem of efficiently computing tree representations of set families. Assuming the existence of efficient algorithms for solving the Membership and Separation problems, we prove that if a set family satisfies weak closure properties then there exists an efficient algorithm for computing a tree representation of the set family. The running time of the algorithm will mainly depend on the running times of the algorithms for the two basic problems. Our algorithm generalizes several previous results and provides a unified approach to the computation for a large class of decompositions of graphs. We also introduce a decomposition notion for directed graphs which has no undirected analogue. We show that the results of the first part of the paper are applicable to this new decomposition. Finally, we give efficient algorithms for the two basic problems and obtain an O(n3)-time algorithm for computing a tree representation.

Obtaining a Bipartite Graph by Contracting Few Edges

Abstract: We initiate the study of the Bipartite Contraction problem from the perspective of parameterized complexity. In this problem we are given a graph $G$ and an integer $k$, and the task is to determine whether we can obtain a bipartite graph from $G$ by a sequence of at most $k$ edge contractions. Our main result is an $f(k) n^{O(1)}$ time algorithm for Bipartite Contraction. Despite a strong resemblance between Bipartite Contraction and the classical Odd Cycle Transversal (OCT) problem, the methods developed to tackle OCT do not seem to be directly applicable to Bipartite Contraction. Our algorithm is based on a novel combination of the irrelevant vertex technique, introduced by Robertson and Seymour, and the concept of important separators. Both techniques have previously been used as key components of algorithms for fundamental problems in parameterized complexity. However, to the best of our knowledge, this is the first time the two techniques are applied in unison.