Kernelization using structural parameters on sparse graph classes

Gajarský, Jakub; Hlinźný, Petr; Óbdržálek, Jan; Ordyniak, Sebastian; Reidl, Felix; Rossmanith, Peter; Villaamil, Fernando Sánchez; Sikdar, Somnath

doi:10.1016/j.jcss.2016.09.002

Cited by 64 publications

(49 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example all classes of bounded VC-dimension have polynomially bounded neighborhood complexity functions [44,45], while bounded expansion classes have linear and nowhere dense classes have almost linear neighborhood complexity [20]. Proposition 2.5.…”

Section: A General Frameworkmentioning

confidence: 99%

Lossy Kernels for Connected Dominating Set on Sparse Graphs

Eiben¹,

Kumar²,

Mouawad³

et al. 2019

SIAM J. Discrete Math.

View full text Add to dashboard Cite

For α > 1, an α-approximate (bi-)kernel is a polynomial-time algorithm that takes as input an instance (I, k) of a problem Q and outputs an instance (I , k ) (of a problem Q ) of size bounded by a function of k such that, for every c ≥ 1, a c-approximate solution for the new instance can be turned into a (c · α)-approximate solution of the original instance in polynomial time. This framework of lossy kernelization was recently introduced by Lokshtanov et al. We study Connected Dominating Set (and its distance-r variant) parameterized by solution size on sparse graph classes like biclique-free graphs, classes of bounded expansion, and nowhere dense classes. We prove that for every α > 1, Connected Dominating Set admits a polynomial-size α-approximate (bi-)kernel on all the aforementioned classes. Our results are in sharp contrast to the kernelization complexity of Connected Dominating Set, which is known to not admit a polynomial kernel even on 2-degenerate graphs and graphs of bounded expansion, unless NP ⊆ coNP/poly. We complement our results by the following conditional lower bound. We show that if a class C is somewhere dense and closed under taking subgraphs, then for some value of r ∈ N there cannot exist an α-approximate bi-kernel for the (Connected) Distance-r Dominating Set problem on C for any α > 1 (assuming the Gap Exponential Time Hypothesis).

show abstract

Section: A General Frameworkmentioning

confidence: 99%

Lossy Kernels for Connected Dominating Set on Sparse Graphs

Eiben¹,

Kumar²,

Mouawad³

et al. 2019

SIAM J. Discrete Math.

View full text Add to dashboard Cite

show abstract

“…planar graphs or graphs excluding a topological minor), the number of twin classes in V (G) \ X is bounded linearly in t and in nowhere dense classes by t 1+o(1) (cf. Lemma 4.3 and Corollary 4.4 in [13]). Furthermore, if the input graphs stem from a d-degenerate class, the number of twin-classes and thus the number of vertices in the kernel is bounded by O(t d+1 ); a fact that follows easily from the observation that in such a class at most dt vertices in the independent set can have degree more than d. It is therefore natural to ask whether this structural parameterization allows a polynomial kernel in general graphs, a question we answer in the negative.…”

Section: Structural Parameterizations For Metric Dimensionmentioning

confidence: 88%

Alternative parameterizations of Metric Dimension

Gutin

Ramanujan

Reidl

et al. 2020

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

A set of vertices W in a graph G is called resolving if for any two distinct x, y ∈ V (G), there is v ∈ W such that dG(v, x) = dG(v, y), where dG(u, v) denotes the length of a shortest path between u and v in the graph G. The metric dimension md(G) of G is the minimum cardinality of a resolving set. The Metric Dimension problem, i.e. deciding whether md(G) k, is NP-complete even for interval graphs (Foucaud et al., 2017). We study Metric Dimension (for arbitrary graphs) from the lens of parameterized complexity. The problem parameterized by k was proved to be W[2]-hard by Hartung and Nichterlein (2013) and we study the dual parameterization, i.e., the problem of whether md(G) n − k, where n is the order of G. We prove that the dual parameterization admits (a) a kernel with at most 3k 4 vertices and (b) an algorithm of runtime O * (4 k+o(k) ). Hartung and Nichterlein (2013) also observed that Metric Dimension is fixed-parameter tractable when parameterized by the vertex cover number vc(G) of the input graph. We complement this observation by showing that it does not admit a polynomial kernel even when parameterized by vc(G) + k. Our reduction also gives evidence for non-existence of polynomial Turing kernels.

show abstract

“…We believe that our work is another good example of how abstract properties derived from the sparsity of the considered graph class, in particular the ones expressed in Proposition 1, can be used in the kernelization setting for a clean treatment of graph classes with excluded minors, without the need of invoking the decomposition theorem of Robertson and Seymour. Other examples of this approach include [15,20], and we hope that even more will appear in future.…”

Section: Discussionmentioning

confidence: 99%

Linear Kernels for Outbranching Problems in Sparse Digraphs

et al. 2016

View full text Add to dashboard Cite

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 whether there exists an outbranching rooted at r that has at least k leaves, or at least k internal vertices, respectively. Both these problems have been 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 H-minor-free graphs, for any fixed family of graphs H, whereas k-Internal Out-Branching admits a kernel with O(k) vertices on any graph class of bounded expansion.

show abstract

Kernelization using structural parameters on sparse graph classes

Cited by 64 publications

References 38 publications

Lossy Kernels for Connected Dominating Set on Sparse Graphs

Lossy Kernels for Connected Dominating Set on Sparse Graphs

Alternative parameterizations of Metric Dimension

Linear Kernels for Outbranching Problems in Sparse Digraphs

Contact Info

Product

Resources

About