Robert Ganian

On parse trees and Myhill-Nerode-type tools for handling graphs of bounded rank-width

Rank-width is a structural graph measure introduced by Oum and Seymour and aimed at better handling of graphs of bounded clique-width. We propose a formal mathematical framework and tools for easy design of dynamic algorithms running directly on a rank-decomposition of a graph (on contrary to the usual approach which translates a rank-decomposition into a clique-width expression, with a possible exponential jump in the parameter). The main advantage of this framework is a fine control over the runtime dependency on the rank-width parameter. Our new approach is linked to a work of Courcelle and Kante [7] who first proposed algebraic expressions with a so-called bilinear graph product as a better way of handling rank-decompositions, and to a parallel recent research of Bui-Xuan, Telle and Vatshelle.

Are There Any Good Digraph Width Measures

Several different measures for digraph width have appeared in the last few years. However, none of them shares all the nice properties of treewidth: First, being emph{algorithmically useful} i.e. admitting polynomial-time algorithms for all

On Digraph Width Measures in Parameterized Algorithmics

MS1

Twin-Cover: beyond vertex cover in parameterized algorithmics

-definable problems on digraphs of bounded width. And, second, having nice emph{structural properties} i.e. being monotone under taking subdigraphs and some form of arc contractions. As for the former, (undirected)

Digraph width measures in parameterized algorithmics

MS1

Better Algorithms for Satisfiability Problems for Formulas of Bounded Rank-width

seems to be the least common denominator of all reasonably expressive logical languages on digraphs that can speak about the edge/arc relation on the vertex set.The latter property is a necessary condition for a width measure to be characterizable by some version of the cops-and-robber game characterizing the ordinary treewidth. Our main result is that emph{any reasonable} algorithmically useful and structurally nice digraph measure cannot be substantially different from the treewidth of the underlying undirected graph. Moreover, we introduce emph{directed topological minors} and argue that they are the weakest useful notion of minors for digraphs.

Thread Graphs, Linear Rank-Width and Their Algorithmic Applications

In contrast to undirected width measures (such as tree-width or clique-width), which have provided many important algorithmic applications, analogous measures for digraphs such as DAG-width or Kelly-width do not seem so successful. Several recent papers, e.g. those of Kreutzer---Ordyniak, Dankelmann---Gutin---Kim, or Lampis---Kaouri---Mitsou, have given some evidence for this. We support this direction by showing that many quite different problems remain hard even on graph classes that are restricted very beyond simply having small DAG-width. To this end, we introduce new measures K-width and DAG-depth. On the positive side, we also note that taking Kantes directed generalization of rank-width as a parameter makes many problems fixed parameter tractable.

Lower bounds on the complexity of MSO1 model-checking

Parameterized algorithms are a very useful tool for dealing with NP-hard problems on graphs. In this context, vertex cover is used as a powerful parameter for dealing with problems which are hard to solve even on graphs of bounded tree-width. The drawback of vertex cover is that bounding it severely restricts admissible graph classes. We introduce a new parameter called twin-cover and show that it is capable of solving a wide range of hard problems while also being much less restrictive than vertex cover and attaining low values even on dense graphs. n nThe article begins by introducing a new FPT algorithm for Graph Motif on graphs of bounded vertex cover. This is the first algorithm of this kind for Graph Motif. We continue by defining twin-cover and providing some related results and notions. The next section contains a number of new FPT algorithms on graphs of bounded twin-cover, with a special emphasis on solving problems which are hard even on graphs of bounded tree-width. Finally, section five generalizes the recent results of Michael Lampis for MS1 model checking from vertex cover to twin-cover.

A unified approach to polynomial algorithms on graphs of bounded (bi-)rank-width

Abstract In contrast to undirected width measures such as tree-width, which have provided many important algorithmic applications, analogous measures for digraphs such as directed tree-width or DAG-width do not seem so successful. Several recent papers have given some evidence on the negative side. We confirm and consolidate this overall picture by thoroughly and exhaustively studying the complexity of a range of directed problems with respect to various parameters, and by showing that they often remain NP-hard even on graph classes that are restricted very beyond having small DAG-width. On the positive side, it turns out that clique-width (of digraphs) performs much better on virtually all considered problems, from the parameterized complexity point of view.

Algorithmic Applications of Tree-Cut Width

We provide a parameterized algorithm for the propositional model counting problem #SAT, the runtime of which has a single-exponential dependency on the rank-width of the signed graph of a formula. That is, our algorithm runs in time