Solving Problems on Graphs of High Rank-Width

Abstract

A modulator in a graph is a vertex set whose deletion places the considered graph into some specified graph class. The cardinality of a modulator to various graph classes has long been used as a structural parameter which can be exploited to obtain fixed-parameter algorithms for a range of hard problems. Here we investigate what happens when a graph contains a modulator which is large but "well-structured" (in the sense of having bounded rank-width). Can such modulators still be exploited to obtain efficient algorithms? And is it even possible to find such modulators efficiently? We first show that the parameters derived from such well-structured modulators are more powerful for fixed-parameter algorithms than the cardinality of modulators and rank-width itself. Then, we develop a fixed-parameter algorithm for finding such wellstructured modulators to every graph class which can be characterized by a finite set of forbidden induced subgraphs. We proceed by showing how well-structured modulators can be used to obtain efficient parameterized algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use the concept of well-structured modulators to develop an algorithmic meta-theorem for deciding problems expressible in monadic second order logic, and prove that this result is tight in the sense that it cannot be generalized to LinEMSO problems.

Vertex Deletion Parameterized by Elimination Distance and Even Less

“…Our results fall in a recent line of work of using hybrid parameterizations [17,18,32,33,42,43,44,46,47,48,54] which simultaneously capture the connectivity structure of the input instance and properties of its optimal solutions. Several of these works give decomposition algorithms for parameterizations similar to ours; we discuss those which are most relevant.…”

“…Eiben, Ganian, and Szeider [33] introduced the hybrid parameter H-well-structure number of a graph G, which is a hybrid of the vertex-deletion distance and rankwidth of a graph. They give FPT algorithms with unspecified parameter dependence f (k) to compute the corresponding graph decomposition for classes H defined by a finite set of forbidden induced subgraphs, forests, and chordal graphs, and use this to solve Vertex cover parameterized by the H-well-structure number.…”

Measuring what Matters: A Hybrid Approach to Dynamic Programming with Treewidth

“…A subset of the authors previously studied modulator-based parameters that used graph splits and rank-width. The resulting well-structured modulators could either dominate rank-width [19], or (depending on the specific constants used) be used to obtain polynomial kernels for a variety of problems [18]. Since R c -treewidth lies "between" treewidth and rank-width, it is clear that our class of parameters is different from these previously studied ones.…”

Towards exact structural thresholds for parameterized complexity

“…On the algorithmic side, the study of heterogeneous parameterizations has been gaining traction [6,7,22,24,23,32,39], yielding the notions of H-treewidth and H-elimination distance, which is a generalization of treedepth, where H is a class of graphs. In particular, the problem of computing H-elimination distance is receiving considerable attention [1,2,3,25,38].…”