Meta-Kernelization with Structural Parameters

Meta-kernelization theorems are general results that provide polynomial kernels for large classes of parameterized problems. The known meta-kernelization theorems, in particular the results of Bodlaender et al. (FOCS'09) and of Fomin et al. (FOCS'10), apply to optimization problems parameterized by solution size. We present the first meta-kernelization theorems that use a structural parameters of the input and not the solution size. Let C be a graph class. We define the C-cover number of a graph to be a the smallest number of modules the vertex set can be partitioned into, such that each module induces a subgraph that belongs to the class C. We show that each graph problem that can be expressed in Monadic Second Order (MSO) logic has a polynomial kernel with a linear number of vertices when parameterized by the C-cover number for any fixed class C of bounded rank-width (or equivalently, of bounded clique-width, or bounded Boolean width). Many graph problems such as Independent Dominating Set, c-Coloring, and c-Domatic Number are covered by this meta-kernelization result. Our second result applies to MSO expressible optimization problems, such as Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique. We show that these problems admit a polynomial annotated kernel with a linear number of vertices

A New Perspective on Vertex Connectivity

Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex connectivity is much more limited. An essential tool in capturing edge connectivity are edge-disjoint spanning trees. The famous results of Tutte and Nash-Williams show that a graph with edge connectivity $\lambda$ contains \floor{\lambda/2} edge-disjoint spanning trees. We present connected dominating set (CDS) partition and packing as tools that are analogous to edge-disjoint spanning trees and that help us to better grasp the structure of graphs based on their vertex connectivity. The objective of the CDS partition problem is to partition the nodes of a graph into as many connected dominating sets as possible. The CDS packing problem is the corresponding fractional relaxation, where CDSs are allowed to overlap as long as this is compensated by assigning appropriate weights. CDS partition and CDS packing can be viewed as the counterparts of the well-studied edge-disjoint spanning trees, focusing on vertex disjointedness rather than edge disjointness. We constructively show that every $k$-vertex-connected graph with $n$ nodes has a CDS packing of size $\Omega(k/\log n)$ and a CDS partition of size $\Omega(k/\log^5 n)$. We prove that the $\Omega(k/\log n)$ CDS packing bound is existentially optimal. Using CDS packing, we show that if vertices of a $k$-vertex-connected graph are independently sampled with probability $p$, then the graph induced by the sampled vertices has vertex connectivity $\tilde{\Omega}(kp^2)$. Moreover, using our $\Omega(k/\log n)$ CDS packing, we get a store-and-forward broadcast algorithm with optimal throughput in the networking model where in each round, each node can send one bounded-size message to all its neighbors

10211 Abstracts Collection -- Flexible Network Design

From Monday 24.05.2010---Friday 28.05.2010, the Dagstuhl Seminar 10211 ``Flexible Network Design \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available