arXiv subjects

Sebastian Siebertz

Publications and source records attributed to Sebastian Siebertz.

At least 73 records · Page 4Linked to original sources

arXiv2019

Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-Wideness

The notions of bounded expansion and nowhere denseness not only offer robust and general definitions of uniform sparseness of graphs, they also describe the tractability boundary for several important algorithmic questions. In this paper we study two structural properties of these graph classes that are of particular importance in this context, namely the property of having bounded generalized coloring numbers and the property of being uniformly quasi-wide. We provide experimental evaluations of several algorithms that approximate these parameters on real-world graphs. On the theoretical side, we provide a new algorithm for uniform quasi-wideness with polynomial size guarantees in graph classes of bounded expansion and show a lower bound indicating that the guarantees of this algorithm are close to optimal in graph classes with fixed excluded minor.

cs.DM↗

arXiv2019

Nowhere dense graph classes and algorithmic applications. A tutorial at Highlights of Logic, Games and Automata 2019

The notion of nowhere dense graph classes was introduced by Nešetřil and Ossona de Mendez and provides a robust concept of uniform sparseness of graph classes. Nowhere dense classes generalize many familiar classes of sparse graphs such as classes that exclude a fixed graph as a minor or topological minor. They admit several seemingly unrelated natural characterizations that lead to strong algorithmic applications. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over these classes. These notes, prepared for a tutorial at Highlights of Logic, Games and Automata 2019, are a brief introduction to the theory of nowhere denseness, driven by algorithmic applications.

cs.DM↗

arXiv2019

Classes of graphs with low complexity: the case of classes with bounded linear rankwidth

Classes with bounded rankwidth are MSO-transductions of trees and classes with bounded linear rankwidth are MSO-transductions of paths -- a result that shows a strong link between the properties of these graph classes considered from the point of view of structural graph theory and from the point of view of finite model theory. We take both views on classes with bounded linear rankwidth and prove structural and model theoretic properties of these classes. The structural results we obtain are the following. 1) The number of unlabeled graphs of order $n$ with linear rank-width at most~$r$ is at most $\bigl[(r/2)!\,2^{\binom{r}{2}}3^{r+2}\bigr]^n$. 2) Graphs with linear rankwidth at most $r$ are linearly $χ$-bounded. Actually, they have bounded $c$-chromatic number, meaning that they can be colored with $f(r)$ colors, each color inducing a cograph. 3) To the contrary, based on a Ramsey-like argument, we prove for every proper hereditary family $F$ of graphs (like cographs) that there is a class with bounded rankwidth that does not have the property that graphs in it can be colored by a bounded number of colors, each inducing a subgraph in $F$. From the model theoretical side we obtain the following results: 1) A direct short proof that graphs with linear rankwidth at most $r$ are first-order transductions of linear orders. This result could also be derived from Colcombet's theorem on first-order transduction of linear orders and the equivalence of linear rankwidth with linear cliquewidth. 2) For a class $C$ with bounded linear rankwidth the following conditions are equivalent: a) $C$ is stable, b) $C$ excludes some half-graph as a semi-induced subgraph, c) $C$ is a first-order transduction of a class with bounded pathwidth. These results open the perspective to study classes admitting low linear rankwidth covers.

cs.LO↗

arXiv2019

On low rank-width colorings

We introduce the concept of low rank-width colorings, generalising the notion of low tree-depth colorings introduced by Nešetřil and Ossona de Mendez in [Grad and classes with bounded expansion I. Decompositions. EJC, 2008]. We say that a class $\mathcal{C}$ of graphs admits low rank-width colourings if there exist functions $N\colon \mathbb{N}\rightarrow\mathbb{N}$ and $Q\colon \mathbb{N}\rightarrow\mathbb{N}$ such that for all $p\in \mathbb{N}$, every graph $G\in \mathcal{C}$ can be vertex colored with at most $N(p)$ colors such that the union of any $i\leq p$ color classes induces a subgraph of rank-width at most $Q(i)$. Graph classes admitting low rank-width colorings strictly generalize graph classes admitting low tree-depth colorings and graph classes of bounded rank-width. We prove that for every graph class $\mathcal{C}$ of bounded expansion and every positive integer $r$, the class $\{G^r\colon G\in \mathcal{C}\}$ of $r$th powers of graphs from $\mathcal{C}$, as well as the classes of unit interval graphs and bipartite permutation graphs admit low rank-width colorings. All of these classes have unbounded rank-width and do not admit low tree-depth colorings. We also show that the classes of interval graphs and permutation graphs do not admit low rank-width colorings. As interesting side properties, we prove that every graph class admitting low rank-width colorings has the Erdős-Hajnal property and is $χ$-bounded.

cs.DS↗

arXiv2019

Distributed Dominating Set Approximations beyond Planar Graphs

The Minimum Dominating Set (MDS) problem is one of the most fundamental and challenging problems in distributed computing. While it is well-known that minimum dominating sets cannot be approximated locally on general graphs, over the last years, there has been much progress on computing local approximations on sparse graphs, and in particular planar graphs. In this paper we study distributed and deterministic MDS approximation algorithms for graph classes beyond planar graphs. In particular, we show that existing approximation bounds for planar graphs can be lifted to bounded genus graphs, and present (1) a local constant-time, constant-factor MDS approximation algorithm and (2) a local $\mathcal{O}(\log^*{n})$-time approximation scheme. Our main technical contribution is a new analysis of a slightly modified variant of an existing algorithm by Lenzen et al. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on direct topological arguments.

cs.DC↗

arXiv2019

Greedy domination on biclique-free graphs

The greedy algorithm for approximating dominating sets is a simple method that is known to compute an $(\ln n+1)$-approximation of a minimum dominating set on any graph with $n$ vertices. We show that a small modification of the greedy algorithm can be used to compute an $O(t^2\cdot \ln k)$-approximation, where~$k$ is the size of a minimum dominating set, on graphs that exclude the complete bipartite graph $K_{t,t}$ as a subgraph.

cs.DM↗

arXiv2018

Model-Checking on Ordered Structures

We study the model-checking problem for first- and monadic second-order logic on finite relational structures. The problem of verifying whether a formula of these logics is true on a given structure is considered intractable in general, but it does become tractable on interesting classes of structures, such as on classes whose Gaifman graphs have bounded treewidth. In this paper we continue this line of research and study model-checking for first- and monadic second-order logic in the presence of an ordering on the input structure. We do so in two settings: the general ordered case, where the input structures are equipped with a fixed order or successor relation, and the order invariant case, where the formulas may resort to an ordering, but their truth must be independent of the particular choice of order. In the first setting we show very strong intractability results for most interesting classes of structures. In contrast, in the order invariant case we obtain tractability results for order-invariant monadic second-order formulas on the same classes of graphs as in the unordered case. For first-order logic, we obtain tractability of successor-invariant formulas on classes whose Gaifman graphs have bounded expansion. Furthermore, we show that model-checking for order-invariant first-order formulas is tractable on coloured posets of bounded width.

cs.LO↗

arXiv2018

Progressive Algorithms for Domination and Independence

We consider a generic algorithmic paradigm that we call progressive exploration, which can be used to develop simple and efficient parameterized graph algorithms. We identify two model-theoretic properties that lead to efficient progressive algorithms, namely variants of the Helly property and stability. We demonstrate our approach by giving linear-time fixed-parameter algorithms for the distance-r dominating set problem (parameterized by the solution size) in a wide variety of restricted graph classes, such as powers of nowhere dense classes, map graphs, and (for $r=1$) biclique-free graphs. Similarly, for the distance-r independent set problem the technique can be used to give a linear-time fixed-parameter algorithm on any nowhere dense class. Despite the simplicity of the method, in several cases our results extend known boundaries of tractability for the considered problems and improve the best known running times.

cs.LO↗

arXiv2018

Colouring and Covering Nowhere Dense Graphs

It was shown by Grohe et al. that nowhere dense classes of graphs admit sparse neighbourhood covers of small degree. We show that a monotone graph class admits sparse neighbourhood covers if and only if it is nowhere dense. The existence of such covers for nowhere dense classes is established through bounds on so-called weak colouring numbers. The core results of this paper are various lower and upper bounds on the weak colouring numbers and other, closely related generalised colouring numbers. We prove tight bounds for these numbers on graphs of bounded tree width. We clarify and tighten the relation between the expansion (in the sense of "bounded expansion" as defined by Nesetril and Ossona de Mendez) and the various generalised colouring numbers. These upper bounds are complemented by new, stronger exponential lower bounds on the generalised colouring numbers. Finally, we show that computing weak r-colouring numbers is NP-complete for all r>2.

math.CO↗

arXiv2018

First-order interpretations of bounded expansion classes

The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes of graphs with structurally bounded expansion, defined as first-order interpretations of classes of bounded expansion. As a first step towards their algorithmic treatment, we provide their characterization analogous to the characterization of classes of bounded expansion via low treedepth decompositions, replacing treedepth by its dense analogue called shrubdepth.

cs.DM↗

arXiv2018

Reconfiguration on nowhere dense graph classes

Let $\mathcal{Q}$ be a vertex subset problem on graphs. In a reconfiguration variant of $\mathcal{Q}$ we are given a graph $G$ and two feasible solutions $S_s, S_t\subseteq V(G)$ of $\mathcal{Q}$ with $|S_s|=|S_t|=k$. The problem is to determine whether there exists a sequence $S_1,\ldots,S_n$ of feasible solutions, where $S_1=S_s$, $S_n=S_t$, $|S_i|\leq k\pm 1$, and each $S_{i+1}$ results from $S_i$, $1\leq i 0$ we even obtain a kernel of almost linear size $\mathcal{O}(k^{1+ε})$. We then prove that if a class $\mathcal{C}$ is somewhere dense and closed under taking subgraphs, then for some value of $r\geq 1$ the reconfiguration variants of the above problems on $\mathcal{C}$ are $\mathsf{W}[1]$-hard (and in particular we cannot expect the existence of kernelization algorithms). Hence our results show that the limit of tractability for the reconfiguration variants of the distance-$r$ independent set problem and distance-$r$ dominating set problem on subgraph closed graph classes lies exactly on the boundary between nowhere denseness and somewhere denseness.

cs.DM↗

arXiv2018

Polynomial Kernels and Wideness Properties of Nowhere Dense Graph Classes

Nowhere dense classes of graphs are very general classes of uniformly sparse graphs with several seemingly unrelated characterisations. From an algorithmic perspective, a characterisation of these classes in terms of uniform quasi-wideness, a concept originating in finite model theory, has proved to be particularly useful. Uniform quasi-wideness is used in many fpt-algorithms on nowhere dense classes. However, the existing constructions showing the equivalence of nowhere denseness and uniform quasi-wideness imply a non-elementary blow up in the parameter dependence of the fpt-algorithms, making them infeasible in practice. As a first main result of this paper, we use tools from logic, in particular from a subfield of model theory known as stability theory, to establish polynomial bounds for the equivalence of nowhere denseness and uniform quasi-wideness. A powerful method in parameterized complexity theory is to compute a problem kernel in a pre-computation step, that is, to reduce the input instance in polynomial time to a sub-instance of size bounded in the parameter only (independently of the input graph size). Our new tools allow us to obtain for every fixed value of $r$ a polynomial kernel for the distance-$r$ dominating set problem on nowhere dense classes of graphs. This result is particularly interesting, as it implies that for every class $\mathcal{C}$ of graphs which is closed under subgraphs, the distance-$r$ dominating set problem admits a kernel on $\mathcal{C}$ for every value of $r$ if, and only if, it admits a polynomial kernel for every value of $r$ (under the standard assumption of parameterized complexity theory that $\mathrm{FPT} \neq W[2]$).

cs.DM↗

arXiv2018

Distributed Domination on Graph Classes of Bounded Expansion

We provide a new constant factor approximation algorithm for the (connected) distance-$r$ dominating set problem on graph classes of bounded expansion. Classes of bounded expansion include many familiar classes of sparse graphs such as planar graphs and graphs with excluded (topological) minors, and notably, these classes form the most general subgraph closed classes of graphs for which a sequential constant factor approximation algorithm for the distance-$r$ dominating set problem is currently known. Our algorithm can be implemented in the \congestbc model of distributed computing and uses $\mathcal{O}(r^2 \log n)$ communication rounds. Our techniques, which may be of independent interest, are based on a distributed computation of sparse neighborhood covers of small radius on bounded expansion classes. We show how to compute an $r$-neighborhood cover of radius~$2r$ and overlap $f(r)$ on every class of bounded expansion in $\mathcal{O}(r^2 \log n)$ communication rounds for some function~$f$.% in the $\mathcal{CONGEST}_{\mathrm{BC}}$ model. Finally, we show how to use the greater power of the $\mathcal{LOCAL}$ model to turn any distance-$r$ dominating set into a constantly larger connected distance-$r$ dominating set in $3r+1$ rounds on any class of bounded expansion. Combining this algorithm, e.g., with the constant factor approximation algorithm for dominating sets on planar graphs of Lenzen et al.\ gives a constant factor approximation algorithm for connected dominating sets on planar graphs in a constant number of rounds in the $\mathcal{LOCAL}$ model, where the approximation ratio is only $6$ times larger than that of Lenzen et al.'s algorithm.

cs.DC↗

arXiv2018

Parameterized circuit complexity of model checking first-order logic on sparse structures

We prove that for every class $C$ of graphs with effectively bounded expansion, given a first-order sentence $φ$ and an $n$-element structure $\mathbb{A}$ whose Gaifman graph belongs to $C$, the question whether $φ$ holds in $\mathbb{A}$ can be decided by a family of AC-circuits of size $f(φ)\cdot n^c$ and depth $f(φ)+c\log n$, where $f$ is a computable function and $c$ is a universal constant. This places the model-checking problem for classes of bounded expansion in the parameterized circuit complexity class $para-AC^1$. On the route to our result we prove that the basic decomposition toolbox for classes of bounded expansion, including orderings with bounded weak coloring numbers and low treedepth decompositions, can be computed in $para-AC^1$.

cs.DM↗

arXiv2018

Lossy Kernels for Connected Dominating Set on Sparse Graphs

For $α> 1$, an $α$-approximate (bi-)kernel is a polynomial-time algorithm that takes as input an instance $(I, k)$ of a problem $\mathcal{Q}$ and outputs an instance $(I',k')$ (of a problem $\mathcal{Q}'$) of size bounded by a function of $k$ such that, for every $c\geq 1$, a $c$-approximate solution for the new instance can be turned into a $(c\cdotα)$-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 $\textsf{NP} \subseteq \textsf{coNP/poly}$. We complement our results by the following conditional lower bound. We show that if a class $\mathcal{C}$ is somewhere dense and closed under taking subgraphs, then for some value of $r\in\mathbb{N}$ there cannot exist an $α$-approximate bi-kernel for the (Connected) Distance-$r$ Dominating Set problem on $\mathcal{C}$ for any $α>1$ (assuming the Gap Exponential Time Hypothesis).

cs.DS↗

arXiv2017

On the number of types in sparse graphs

We prove that for every class of graphs $\mathcal{C}$ which is nowhere dense, as defined by Nesetril and Ossona de Mendez, and for every first order formula $ϕ(\bar x,\bar y)$, whenever one draws a graph $G\in \mathcal{C}$ and a subset of its nodes $A$, the number of subsets of $A^{|\bar y|}$ which are of the form $\{\bar v\in A^{|\bar y|}\, \colon\, G\modelsϕ(\bar u,\bar v)\}$ for some valuation $\bar u$ of $\bar x$ in $G$ is bounded by $\mathcal{O}(|A|^{|\bar x|+ε})$, for every $ε>0$. This provides optimal bounds on the VC-density of first-order definable set systems in nowhere dense graph classes. We also give two new proofs of upper bounds on quantities in nowhere dense classes which are relevant for their logical treatment. Firstly, we provide a new proof of the fact that nowhere dense classes are uniformly quasi-wide, implying explicit, polynomial upper bounds on the functions relating the two notions. Secondly, we give a new combinatorial proof of the result of Adler and Adler stating that every nowhere dense class of graphs is stable. In contrast to the previous proofs of the above results, our proofs are completely finitistic and constructive, and yield explicit and computable upper bounds on quantities related to uniform quasi-wideness (margins) and stability (ladder indices).

cs.DM↗

arXiv2017

Lossy kernels for connected distance-$r$ domination on nowhere dense graph classes

For $α\colon\mathbb{N}\rightarrow\mathbb{R}$, 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 $c\geq 1$, a $c$-approximate solution for the new instance can be turned into a $c\cdotα(k)$-approximate solution of the original instance in polynomial time. This framework of \emph{lossy kernelization} was recently introduced by Lokshtanov et al. We prove that for every nowhere dense class of graphs, every $α>1$ and $r\in\mathbb{N}$ there exists a polynomial $p$ (whose degree depends only on $r$ while its coefficients depend on $α$) such that the connected distance-$r$ dominating set problem with parameter $k$ admits an $α$-approximate bi-kernel of size $p(k)$. Furthermore, we show that this result cannot be extended to more general classes of graphs which are closed under taking subgraphs by showing that if a class $C$ is somewhere dense and closed under taking subgraphs, then for some value of $r\in\mathbb{N}$ there cannot exist an $α$-approximate bi-kernel for the (connected) distance-$r$ dominating set problem on $C$ for any function $α\colon\mathbb{N}\rightarrow\mathbb{R}$ (assuming the Gap Exponential Time Hypothesis).

cs.DM↗

arXiv2017

Algorithmic Properties of Sparse Digraphs

The notions of bounded expansion and nowhere denseness have been applied very successfully in algorithmic graph theory. We study the corresponding notions of directed bounded expansion and nowhere crownfulness on directed graphs. We show that many of the algorithmic tools that were developed for undirected bounded expansion classes can, with some care, also be applied in their directed counterparts, and thereby we highlight a rich algorithmic structure theory of directed bounded expansion classes. More specifically, we show that the directed Steiner tree problem is fixed-parameter tractable on any class of directed bounded expansion parameterized by the number $k$ of non-terminals plus the maximal diameter $s$ of a strongly connected component in the subgraph induced by the terminals. Our result strongly generalizes a result of Jones et al., who proved that the problem is fixed parameter tractable on digraphs of bounded degeneracy if the set of terminals is required to be acyclic. We furthermore prove that for every integer $r\geq 1$, the distance-$r$ dominating set problem can be approximated up to a factor $O(\log k)$ and the connected distance-$r$ dominating set problem can be approximated up to a factor $O(k\cdot \log k)$ on any class of directed bounded expansion, where $k$ denotes the size of an optimal solution. If furthermore, the class is nowhere crownful, we are able to compute a polynomial kernel for distance-$r$ dominating sets. Polynomial kernels for this problem were not known to exist on any other existing digraph measure for sparse classes.

cs.DM↗