arXiv subjects

Sebastian Siebertz

Publications and source records attributed to Sebastian Siebertz.

80 records · Page 5Linked to original sources

arXiv2017

Model-Checking for Successor-Invariant First-Order Formulas on Graph Classes of Bounded Expansion

A successor-invariant first-order formula is a formula that has access to an auxiliary successor relation on a structure's universe, but the model relation is independent of the particular interpretation of this relation. It is well known that successor-invariant formulas are more expressive on finite structures than plain first-order formulas without a successor relation. This naturally raises the question whether this increase in expressive power comes at an extra cost to solve the model-checking problem, that is, the problem to decide whether a given structure together with some (and hence every) successor relation is a model of a given formula. It was shown earlier that adding successor-invariance to first-order logic essentially comes at no extra cost for the model-checking problem on classes of finite structures whose underlying Gaifman graph is planar [Engelmann et al., 2012], excludes a fixed minor [Eickmeyer et al., 2013] or a fixed topological minor [Eickmeyer and Kawarabayashi, 2016; Kreutzer et al., 2016]. In this work we show that the model-checking problem for successor-invariant formulas is fixed-parameter tractable on any class of finite structures whose underlying Gaifman graphs form a class of bounded expansion. Our result generalises all earlier results and comes close to the best tractability results on nowhere dense classes of graphs currently known for plain first-order logic.

cs.LO↗

arXiv2016

Neighborhood complexity and kernelization for nowhere dense classes of graphs

We prove that whenever $G$ is a graph from a nowhere dense graph class $\mathcal{C}$, and $A$ is a subset of vertices of $G$, then the number of subsets of $A$ that are realized as intersections of $A$ with $r$-neighborhoods of vertices of $G$ is at most $f(r,ε)\cdot |A|^{1+ε}$, where $r$ is any positive integer, $ε$ is any positive real, and $f$ is a function that depends only on the class $\mathcal{C}$. This yields a characterization of nowhere dense classes of graphs in terms of neighborhood complexity, which answers a question posed by Reidl et al. As an algorithmic application of the above result, we show that for every fixed $r$, the parameterized Distance-$r$ Dominating Set problem admits an almost linear kernel on any nowhere dense graph class. This proves a conjecture posed by Drange et al., and shows that the limit of parameterized tractability of Distance-$r$ Dominating Set on subgraph-closed graph classes lies exactly on the boundary between nowhere denseness and somewhere denseness.

cs.DM↗

arXiv2016

The Generalised Colouring Numbers on Classes of Bounded Expansion

The generalised colouring numbers $\mathrm{adm}_r(G)$, $\mathrm{col}_r(G)$, and $\mathrm{wcol}_r(G)$ were introduced by Kierstead and Yang as generalisations of the usual colouring number, also known as the degeneracy of a graph, and have since then found important applications in the theory of bounded expansion and nowhere dense classes of graphs, introduced by Nešetřil and Ossona de Mendez. In this paper, we study the relation of the colouring numbers with two other measures that characterise nowhere dense classes of graphs, namely with uniform quasi-wideness, studied first by Dawar et al. in the context of preservation theorems for first-order logic, and with the splitter game, introduced by Grohe et al. We show that every graph excluding a fixed topological minor admits a universal order, that is, one order witnessing that the colouring numbers are small for every value of $r$. Finally, we use our construction of such orders to give a new proof of a result of Eickmeyer and Kawarabayashi, showing that the model-checking problem for successor-invariant first-order formulas is fixed-parameter tractable on classes of graphs with excluded topological minors.

cs.DM↗

arXiv2016

A local constant factor approximation for the minimum dominating set problem on bounded genus graphs

The Minimum Dominating Set (MDS) problem is not only one of the most fundamental problems in distributed computing, it is also one of the most challenging ones. While it is well-known that minimum dominating sets cannot be approximated locally on general graphs, over the last years, several breakthroughs have been made on computing local approximations on sparse graphs. This paper presents a deterministic and local constant factor approximation for minimum dominating sets on bounded genus graphs, a very large family of sparse graphs. Our main technical contribution is a new analysis of a slightly modified, first-order definable variant of an existing algorithm by Lenzen et al. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on any topological arguments. We believe that our techniques can be useful for the study of local problems on sparse graphs beyond the scope of this paper.

cs.DC↗

arXiv2015

Kernelization and Sparseness: the case of Dominating Set

We prove that for every positive integer $r$ and for every graph class $\mathcal G$ of bounded expansion, the $r$-Dominating Set problem admits a linear kernel on graphs from $\mathcal G$. Moreover, when $\mathcal G$ is only assumed to be nowhere dense, then we give an almost linear kernel on $\mathcal G$ for the classic Dominating Set problem, i.e., for the case $r=1$. These results generalize a line of previous research on finding linear kernels for Dominating Set and $r$-Dominating Set. However, the approach taken in this work, which is based on the theory of sparse graphs, is radically different and conceptually much simpler than the previous approaches. We complement our findings by showing that for the closely related Connected Dominating Set problem, the existence of such kernelization algorithms is unlikely, even though the problem is known to admit a linear kernel on $H$-topological-minor-free graphs. Also, we prove that for any somewhere dense class $\mathcal G$, there is some $r$ for which $r$-Dominating Set is W[$2$]-hard on $\mathcal G$. Thus, our results fall short of proving a sharp dichotomy for the parameterized complexity of $r$-Dominating Set on subgraph-monotone graph classes: we conjecture that the border of tractability lies exactly between nowhere dense and somewhere dense graph classes.

cs.DS↗

arXiv2014

Directed Width Measures and Monotonicity of Directed Graph Searching

We consider generalisations of tree width to directed graphs, that attracted much attention in the last fifteen years. About their relative strength with respect to "bounded width in one measure implies bounded width in the other" many problems remain unsolved. Only some results separating directed width measures are known. We give an almost complete picture of this relation. For this, we consider the cops and robber games characterising DAG-width and directed tree width (up to a constant factor). For DAG-width games, it is an open question whether the robber-monotonicity cost (the difference between the minimal numbers of cops capturing the robber in the general and in the monotone case) can be bounded by any function. Examples show that this function (if it exists) is at least $f(k) > 4k/3$ (Kreutzer, Ordyniak 2008). We approach a solution by defining weak monotonicity and showing that if $k$ cops win weakly monotonically, then $O(k^2)$ cops win monotonically. It follows that bounded Kelly-width implies bounded DAG-width, which has been open since the definition of Kelly-width by Hunter and Kreutzer in 2008. For directed tree width games we show that, unexpectedly, the cop-monotonicity cost (no cop revisits any vertex) is not bounded by any function. This separates directed tree width from D-width defined by Safari in 2005, refuting his conjecture.

cs.DM↗

arXiv2014

Deciding first-order properties of nowhere dense graphs

Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez, form a large variety of classes of "sparse graphs" including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion. We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes. At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption). As a by-product, we give an algorithmic construction of sparse neighbourhood covers for nowhere dense graphs. This extends and improves previous constructions of neighbourhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those. Our proofs are based on a new game-theoretic characterisation of nowhere dense graphs that allows for a recursive version of locality-based algorithms on these classes. On the logical side, we prove a "rank-preserving" version of Gaifman's locality theorem.

cs.LO↗

arXiv2013

Vertex Disjoint Path in Upward Planar Graphs

The $k$-vertex disjoint paths problem is one of the most studied problems in algorithmic graph theory. In 1994, Schrijver proved that the problem can be solved in polynomial time for every fixed $k$ when restricted to the class of planar digraphs and it was a long standing open question whether it is fixed-parameter tractable (with respect to parameter $k$) on this restricted class. Only recently, \cite{CMPP}.\ achieved a major breakthrough and answered the question positively. Despite the importance of this result (and the brilliance of their proof), it is of rather theoretical importance. Their proof technique is both technically extremely involved and also has at least double exponential parameter dependence. Thus, it seems unrealistic that the algorithm could actually be implemented. In this paper, therefore, we study a smaller class of planar digraphs, the class of upward planar digraphs, a well studied class of planar graphs which can be drawn in a plane such that all edges are drawn upwards. We show that on the class of upward planar digraphs the problem (i) remains NP-complete and (ii) the problem is fixed-parameter tractable. While membership in FPT follows immediately from \cite{CMPP}'s general result, our algorithm has only single exponential parameter dependency compared to the double exponential parameter dependence for general planar digraphs. Furthermore, our algorithm can easily be implemented, in contrast to the algorithm in \cite{CMPP}.

cs.CC↗