Arrangement Problems Parameterized by Neighbourhood Diversity

We show that the arrangement problems BANDWIDTH, DISTORTION and IMBALANCE is FPT when parameterized by neighbourhood diversity, and establish some structural results for CUTWIDTH parameterized by neighbourhood diversity.Masteroppgave i informatikkINF39

Kernel Bounds for Structural Parameterizations of Pathwidth

Assuming the AND-distillation conjecture, the Pathwidth problem of determining whether a given graph G has pathwidth at most k admits no polynomial kernelization with respect to k. The present work studies the existence of polynomial kernels for Pathwidth with respect to other, structural, parameters. Our main result is that, unless NP is in coNP/poly, Pathwidth admits no polynomial kernelization even when parameterized by the vertex deletion distance to a clique, by giving a cross-composition from Cutwidth. The cross-composition works also for Treewidth, improving over previous lower bounds by the present authors. For Pathwidth, our result rules out polynomial kernels with respect to the distance to various classes of polynomial-time solvable inputs, like interval or cluster graphs. This leads to the question whether there are nontrivial structural parameters for which Pathwidth does admit a polynomial kernelization. To answer this, we give a collection of graph reduction rules that are safe for Pathwidth. We analyze the success of these results and obtain polynomial kernelizations with respect to the following parameters: the size of a vertex cover of the graph, the vertex deletion distance to a graph where each connected component is a star, and the vertex deletion distance to a graph where each connected component has at most c vertices.Comment: This paper contains the proofs omitted from the extended abstract published in the proceedings of Algorithm Theory - SWAT 2012 - 13th Scandinavian Symposium and Workshops, Helsinki, Finland, July 4-6, 201

Related Orderings of AT-Free Graphs

An ordering of a graph G is a bijection of V(G) to {1, . . . , |V(G)|}. In this thesis, we consider the complexity of two types of ordering problems. The first type of problem we consider aims at minimizing objective functions related to an ordering of the graph. We consider the problems Cutwidth, Imbalance, and Optimal Linear Arrangement. We also consider a problem of another type: S-End-Vertex, where S is one of the following search algorithms: breadth-first search (BFS), lexicographic breadth-first search (LBFS), depth-first search (DFS), and maximal neighbourhood search (MNS). This problem asks if a specified vertex can be the last vertex in an ordering generated by S. We show that, for each type of problem, orderings for one problem may be related to orderings for another problem of that type. We show that there is always a cutwidth-minimal ordering where equivalence classes of true twins are grouped for any graph, where true twins are vertices with the same closed neighbourhood. This enables a fixed-parameter tractable (FPT) algorithm for Cutwidth on graphs parameterized by the edge clique cover number of the graph and a new parameter, the restricted twin cover number of the graph. The restricted twin cover number of the graph generalizes the vertex cover number of a graph, and is the smallest value k ≥ 0 such that there is a twin cover of the graph T and k−|T| non-trivial components of G−T. We show that there is also always an imbalance-minimal ordering where equivalence classes of true twins are grouped for any graph. We show a polynomial time algorithm for this problem on superfragile graphs and subsets of proper interval graphs, both subsets of AT-free graphs. An asteroidal triple (AT) is a triple of independent vertices x, y, z such that between every pair of vertices in the triple, there is a path that does not intersect the closed neighbourhood of the third. A graph without an asteroidal triple is said to be AT-free. We also provide closed formulas for Imbalance on some small graph classes. In the FPT setting, we improve algorithms for Imbalance parameterized by the vertex cover number of the input graph and show that the problem does not have a polynomially sized kernel for the same parameter number unless NP ⊆ coNP/poly. We show that Optimal Linear Arrangement also has a polynomial algorithm for superfragile graphs and an FPT algorithm with respect to the restricted twin cover number. Finally, we consider S-End-Vertex, for BFS, LBFS, DFS, and MNS. We perform the first systematic study of the problem on bipartite permutation graphs, a subset of AT-free graphs. We show that for BFS and MNS, the problem has a polynomial time solution. We improve previous results for LBFS, obtaining a linear time algorithm. For DFS, we establish a linear time algorithm. All the results follow from the linear structure of bipartite permutation graphs

Order-Related Problems Parameterized by Width

In the main body of this thesis, we study two different order theoretic problems. The first problem, called Completion of an Ordering, asks to extend a given finite partial order to a complete linear order while respecting some weight constraints. The second problem is an order reconfiguration problem under width constraints. While the Completion of an Ordering problem is NP-complete, we show that it lies in FPT when parameterized by the interval width of ρ. This ordering problem can be used to model several ordering problems stemming from diverse application areas, such as graph drawing, computational social choice, and computer memory management. Each application yields a special partial order ρ. We also relate the interval width of ρ to parameterizations for these problems that have been studied earlier in the context of these applications, sometimes improving on parameterized algorithms that have been developed for these parameterizations before. This approach also gives some practical sub-exponential time algorithms for ordering problems. In our second main result, we combine our parameterized approach with the paradigm of solution diversity. The idea of solution diversity is that instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms that output a small set of sufficiently good solutions that are sufficiently diverse from one another. In this way, the user has the opportunity to choose the solution that is most appropriate to the context at hand. It also displays the richness of the solution space. There, we show that the considered diversity version of the Completion of an Ordering problem is fixed-parameter tractable with respect to natural paramaters that capture the notion of diversity and the notion of sufficiently good solutions. We apply this algorithm in the study of the Kemeny Rank Aggregation class of problems, a well-studied class of problems lying in the intersection of order theory and social choice theory. Up to this point, we have been looking at problems where the goal is to find an optimal solution or a diverse set of good solutions. In the last part, we shift our focus from finding solutions to studying the solution space of a problem. There we consider the following order reconfiguration problem: Given a graph G together with linear orders τ and τ ′ of the vertices of G, can one transform τ into τ ′ by a sequence of swaps of adjacent elements in such a way that at each time step the resulting linear order has cutwidth (pathwidth) at most w? We show that this problem always has an affirmative answer when the input linear orders τ and τ ′ have cutwidth (pathwidth) at most w/2. Using this result, we establish a connection between two apparently unrelated problems: the reachability problem for two-letter string rewriting systems and the graph isomorphism problem for graphs of bounded cutwidth. This opens an avenue for the study of the famous graph isomorphism problem using techniques from term rewriting theory. In addition to the main part of this work, we present results on two unrelated problems, namely on the Steiner Tree problem and on the Intersection Non-emptiness problem from automata theory.Doktorgradsavhandlin

Cutwidth: obstructions and algorithmic aspects

Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most $k$ are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most $k$. We prove that every minimal immersion obstruction for cutwidth at most $k$ has size at most $2^{O(k^3\log k)}$. As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time $2^{O(k^2\log k)}\cdot n$, where $k$ is the optimum width and $n$ is the number of vertices. While being slower by a $\log k$-factor in the exponent than the fastest known algorithm, given by Thilikos, Bodlaender, and Serna in [Cutwidth I: A linear time fixed parameter algorithm, J. Algorithms, 56(1):1--24, 2005] and [Cutwidth II: Algorithms for partial $w$-trees of bounded degree, J. Algorithms, 56(1):25--49, 2005], our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts

Vertex deletion into bipartite permutation graphs

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines \u1d4c1₁ and \u1d4c1₂, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, for a given n-vertex graph, whether we can remove at most k vertices to obtain a bipartite permutation graph. This problem is NP-complete by the classical result of Lewis and Yannakakis [John M. Lewis and Mihalis Yannakakis, 1980]. We analyze the structure of the so-called almost bipartite permutation graphs which may contain holes (large induced cycles) in contrast to bipartite permutation graphs. We exploit the structural properties of the shortest hole in a such graph. We use it to obtain an algorithm for the bipartite permutation vertex deletion problem with running time f(k)n^O(1), and also give a polynomial-time 9-approximation algorithm