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

Meta-Kernelization using Well-Structured Modulators

Kernelization investigates exact preprocessing algorithms with performance guarantees. The most prevalent type of parameters used in kernelization is the solution size for optimization problems; however, also structural parameters have been successfully used to obtain polynomial kernels for a wide range of problems. Many of these parameters can be defined as the size of a smallest modulator of the given graph into a fixed graph class (i.e., a set of vertices whose deletion puts the graph into the graph class). Such parameters admit the construction of polynomial kernels even when the solution size is large or not applicable. This work follows up on the research on meta-kernelization frameworks in terms of structural parameters. We develop a class of parameters which are based on a more general view on modulators: instead of size, the parameters employ a combination of rank-width and split decompositions to measure structure inside the modulator. This allows us to lift kernelization results from modulator-size to more general parameters, hence providing smaller kernels. We show (i) how such large but well-structured modulators can be efficiently approximated, (ii) how they can be used to obtain polynomial kernels for any graph problem expressible in Monadic Second Order logic, and (iii) how they allow the extension of previous results in the area of structural meta-kernelization

Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal

The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most $k$ of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a \BigOh(4^kkmn) time algorithm for it, the first algorithm with polynomial runtime of uniform degree for every fixed $k$. It is known that this implies a polynomial-time compression algorithm that turns OCT instances into equivalent instances of size at most \BigOh(4^k), a so-called kernelization. Since then the existence of a polynomial kernel for OCT, i.e., a kernelization with size bounded polynomially in $k$, has turned into one of the main open questions in the study of kernelization. This work provides the first (randomized) polynomial kernelization for OCT. We introduce a novel kernelization approach based on matroid theory, where we encode all relevant information about a problem instance into a matroid with a representation of size polynomial in $k$. For OCT, the matroid is built to allow us to simulate the computation of the iterative compression step of the algorithm of Reed, Smith, and Vetta, applied (for only one round) to an approximate odd cycle transversal which it is aiming to shrink to size $k$. The process is randomized with one-sided error exponentially small in $k$, where the result can contain false positives but no false negatives, and the size guarantee is cubic in the size of the approximate solution. Combined with an \BigOh(\sqrt{\log n})-approximation (Agarwal et al., STOC 2005), we get a reduction of the instance to size \BigOh(k^{4.5}), implying a randomized polynomial kernelization.Comment: Minor changes to agree with SODA 2012 version of the pape

Tight Kernel Bounds for Problems on Graphs with Small Degeneracy

In this paper we consider kernelization for problems on d-degenerate graphs, i.e. graphs such that any subgraph contains a vertex of degree at most $d$. This graph class generalizes many classes of graphs for which effective kernelization is known to exist, e.g. planar graphs, H-minor free graphs, and H-topological-minor free graphs. We show that for several natural problems on d-degenerate graphs the best known kernelization upper bounds are essentially tight.Comment: Full version of ESA 201

Linear kernels for outbranching problems in sparse digraphs

In the $k$-Leaf Out-Branching and $k$-Internal Out-Branching problems we are given a directed graph $D$ with a designated root $r$ and a nonnegative integer $k$. The question is to determine the existence of an outbranching rooted at $r$ that has at least $k$ leaves, or at least $k$ internal vertices, respectively. Both these problems were intensively studied from the points of view of parameterized complexity and kernelization, and in particular for both of them kernels with $O(k^2)$ vertices are known on general graphs. In this work we show that $k$-Leaf Out-Branching admits a kernel with $O(k)$ vertices on $\mathcal{H}$-minor-free graphs, for any fixed family of graphs $\mathcal{H}$, whereas $k$-Internal Out-Branching admits a kernel with $O(k)$ vertices on any graph class of bounded expansion.Comment: Extended abstract accepted for IPEC'15, 27 page

Hierarchies of Inefficient Kernelizability

The framework of Bodlaender et al. (ICALP 2008) and Fortnow and Santhanam (STOC 2008) allows us to exclude the existence of polynomial kernels for a range of problems under reasonable complexity-theoretical assumptions. However, there are also some issues that are not addressed by this framework, including the existence of Turing kernels such as the "kernelization" of Leaf Out Branching(k) into a disjunction over n instances of size poly(k). Observing that Turing kernels are preserved by polynomial parametric transformations, we define a kernelization hardness hierarchy, akin to the M- and W-hierarchy of ordinary parameterized complexity, by the PPT-closure of problems that seem likely to be fundamentally hard for efficient Turing kernelization. We find that several previously considered problems are complete for our fundamental hardness class, including Min Ones d-SAT(k), Binary NDTM Halting(k), Connected Vertex Cover(k), and Clique(k log n), the clique problem parameterized by k log n