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Sebastian Siebertz

Publications and source records attributed to Sebastian Siebertz.

At least 55 records · Page 3Linked to original sources
arXiv2022

A survey on the parameterized complexity of the independent set and (connected) dominating set reconfiguration problems

A graph vertex-subset problem defines which subsets of the vertices of an input graph are feasible solutions. We view a feasible solution as a set of tokens placed on the vertices of the graph. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions of size $k$, whether it is possible to transform one into the other by a sequence of token slides (along edges of the graph) or token jumps (between arbitrary vertices of the graph) such that each intermediate set remains a feasible solution of size $k$. Many algorithmic questions present themselves in the form of reconfiguration problems: Given the description of an initial system state and the description of a target state, is it possible to transform the system from its initial state into the target one while preserving certain properties of the system in the process? Such questions have received a substantial amount of attention under the so-called combinatorial reconfiguration framework. We consider reconfiguration variants of three fundamental underlying graph vertex-subset problems, namely Independent Set, Dominating Set, and Connected Dominating Set. We survey both older and more recent work on the parameterized complexity of all three problems when parameterized by the number of tokens $k$. The emphasis will be on positive results and the most common techniques for the design of fixed-parameter tractable algorithms.

cs.CC↗
arXiv2022

Transducing paths in graph classes with unbounded shrubdepth

Transductions are a general formalism for expressing transformations of graphs (and more generally, of relational structures) in logic. We prove that a graph class $\mathscr{C}$ can be $\mathsf{FO}$-transduced from a class of bounded-height trees (that is, has bounded shrubdepth) if, and only if, from $\mathscr{C}$ one cannot $\mathsf{FO}$-transduce the class of all paths. This establishes one of the three remaining open questions posed by Blumensath and Courcelle about the $\mathsf{MSO}$-transduction quasi-order, even in the stronger form that concerns $\mathsf{FO}$-transductions instead of $\mathsf{MSO}$-transductions. The backbone of our proof is a graph-theoretic statement that says the following: If a graph $G$ excludes a path, the bipartite complement of a path, and a half-graph as semi-induced subgraphs, then the vertex set of $G$ can be partitioned into a bounded number of parts so that every part induces a cograph of bounded height, and every pair of parts semi-induce a bi-cograph of bounded height. This statement may be of independent interest; for instance, it implies that the graphs in question form a class that is linearly $χ$-bounded.

math.CO↗
arXiv2021

Discrepancy and Sparsity

We study the connections between the notions of combinatorial discrepancy and graph degeneracy. In particular, we prove that the maximum discrepancy over all subgraphs $H$ of a graph $G$ of the neighborhood set system of $H$ is sandwiched between $Ω(\log\mathrm{deg}(G))$ and $\mathcal{O}(\mathrm{deg}(G))$, where $\mathrm{deg}(G)$ denotes the degeneracy of $G$. We extend this result to inequalities relating weak coloring numbers and discrepancy of graph powers and deduce a new characterization of bounded expansion classes. Then, we switch to a model theoretical point of view, introduce pointer structures, and study their relations to graph classes with bounded expansion. We deduce that a monotone class of graphs has bounded expansion if and only if all the set systems definable in this class have bounded hereditary discrepancy. Using known bounds on the VC-density of set systems definable in nowhere dense classes we also give a characterization of nowhere dense classes in terms of discrepancy. As consequences of our results, we obtain a corollary on the discrepancy of neighborhood set systems of edge colored graphs, a polynomial-time algorithm to compute $\varepsilon$-approximations of size $\mathcal{O}(1/\varepsilon)$ for set systems definable in bounded expansion classes, an application to clique coloring, and even the non-existence of a quantifier elimination scheme for nowhere dense classes.

cs.DM↗
arXiv2021

Local planar domination revisited

We show how to compute a 20-approximation of a minimum dominating set in a planar graph in a constant number of rounds in the LOCAL model of distributed computing. This improves on the previously best known approximation factor of 52, which was achieved by an elegant and simple algorithm of Lenzen et al. Our algorithm combines ideas from the algorithm of Lenzen et al. with recent work of Czygrinow et al. and Kublenz et al. to reduce to the case of bounded degree graphs, where we can simulate a distributed version of the classical greedy algorithm.

cs.DC↗
arXiv2021

First-Order Logic with Connectivity Operators

First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem, parameterized by solution size. On the other hand, FO cannot express the very simple algorithmic question of whether two vertices are connected. We enrich FO with connectivity predicates that are tailored to express algorithmic graph properties that are commonly studied in parameterized algorithmics. By adding the atomic predicates $conn_k (x, y, z_1 ,\ldots, z_k)$ that hold true in a graph if there exists a path between (the valuations of) $x$ and $y$ after (the valuations of) $z_1,\ldots,z_k$ have been deleted, we obtain separator logic $FO + conn$. We show that separator logic can express many interesting problems such as the feedback vertex set problem and elimination distance problems to first-order definable classes. We then study the limitations of separator logic and prove that it cannot express planarity, and, in particular, not the disjoint paths problem. We obtain the stronger disjoint-paths logic $FO + DP$ by adding the atomic predicates $disjoint-paths_k [(x_1, y_1 ),\ldots , (x_k , y_k )]$ that evaluate to true if there are internally vertex disjoint paths between (the valuations of) $x_i$ and $y_i$ for all $1 \le i \le k$. Disjoint-paths logic can express the disjoint paths problem, the problem of (topological) minor containment, the problem of hitting (topological) minors, and many more. Finally, we compare the expressive power of the new logics with that of transitive closure logics and monadic second-order logic.

cs.LO↗
arXiv2021

Algorithms and data structures for first-order logic with connectivity under vertex failures

We introduce a new data structure for answering connectivity queries in undirected graphs subject to batched vertex failures. Precisely, given any graph G and integer k, we can in fixed-parameter time construct a data structure that can later be used to answer queries of the form: ``are vertices s and t connected via a path that avoids vertices $u_1,..., u_k$?'' in time $2^{2^{O(k)}}$. In the terminology of the literature on data structures, this gives the first deterministic data structure for connectivity under vertex failures where for every fixed number of failures, all operations can be performed in constant time. With the aim to understand the power and the limitations of our new techniques, we prove an algorithmic meta theorem for the recently introduced separator logic, which extends first-order logic with atoms for connectivity under vertex failures. We prove that the model-checking problem for separator logic is fixed-parameter tractable on every class of graphs that exclude a fixed topological minor. We also show a weak converse. This implies that from the point of view of parameterized complexity, under standard complexity assumptions, the frontier of tractability of separator logic is almost exactly delimited by classes excluding a fixed topological minor. The backbone of our proof relies on a decomposition theorem of Cygan et al. [SICOMP '19], which provides a tree decomposition of a given graph into bags that are unbreakable. Crucially, unbreakability allows to reduce separator logic to plain first-order logic within each bag individually. We design our model-checking algorithm using dynamic programming over the tree decomposition, where the transition at each bag amounts to running a suitable model-checking subprocedure for plain first-order logic. This approach is robust enough to provide also efficient enumeration of queries expressed in separator logic.

cs.DS↗
arXiv2021

Structural properties of the first-order transduction quasiorder

Logical transductions provide a very useful tool to encode classes of structures inside other classes of structures. In this paper we study first-order (FO) transductions and the quasiorder they induce on infinite classes of finite graphs. Surprisingly, this quasiorder is very complex, though shaped by the locality properties of first-order logic. This contrasts with the conjectured simplicity of the monadic second order (MSO) transduction quasiorder. We first establish a local normal form for FO transductions, which is of independent interest. Then we prove that the quotient partial order is a bounded distributive join-semilattice, and that the subposet of \emph{additive} classes is also a bounded distributive join-semilattice. The FO transduction quasiorder has a great expressive power, and many well studied class properties can be defined using it. We apply these structural properties to prove, among other results, that FO transductions of the class of paths are exactly perturbations of classes with bounded bandwidth, that the local variants of monadic stability and monadic dependence are equivalent to their (standard) non-local versions, and that the classes with pathwidth at most $k$, for $k\geq 1$ form a strict hierarchy in the FO transduction quasiorder.

math.CO↗
arXiv2021

Parameterized Distributed Complexity Theory: A logical approach

Parameterized complexity theory offers a framework for a refined analysis of hard algorithmic problems. Instead of expressing the running time of an algorithm as a function of the input size only, running times are expressed with respect to one or more parameters of the input instances. In this work we follow the approach of parameterized complexity to provide a framework of parameterized distributed complexity. The central notion of efficiency in parameterized complexity is fixed-parameter tractability and we define the distributed analogue Distributed-FPT (for Distributed in $\{Local, Congest, Congested-Clique\}$) as the class of problems that can be solved in $f(k)$ communication rounds in the Distributed model of distributed computing, where $k$ is the parameter of the problem instance and $f$ is an arbitrary computable function. To classify hardness we introduce three hierarchies. The Distributed-WEFT-hierarchy is defined analogously to the W-hierarchy in parameterized complexity theory via reductions to the weighted circuit satisfiability problem, but it turns out that this definition does not lead to satisfying frameworks for the Local and Congest models. We then follow a logical approach that leads to a more robust theory. We define the levels of the Distributed-W-hierarchy and the Distributed-A-hierarchy that have first-order model-checking problems as their complete problems via suitable reductions.

cs.DC↗
arXiv2021

Recursive Backdoors for SAT

A strong backdoor in a formula $ϕ$ of propositional logic to a tractable class $\mathcal{C}$ of formulas is a set $B$ of variables of $ϕ$ such that every assignment of the variables in $B$ results in a formula from $\mathcal{C}$. Strong backdoors of small size or with a good structure, e.g. with small backdoor treewidth, lead to efficient solutions for the propositional satisfiability problem SAT. In this paper we propose the new notion of recursive backdoors, which is inspired by the observation that in order to solve SAT we can independently recurse into the components that are created by partial assignments of variables. The quality of a recursive backdoor is measured by its recursive backdoor depth. Similar to the concept of backdoor treewidth, recursive backdoors of bounded depth include backdoors of unbounded size that have a certain treelike structure. However, the two concepts are incomparable and our results yield new tractability results for SAT.

cs.DS↗
arXiv2020

Kernelization and approximation of distance-$r$ independent sets on nowhere dense graphs

For a positive integer $r$, a distance-$r$ independent set in an undirected graph $G$ is a set $I\subseteq V(G)$ of vertices pairwise at distance greater than $r$, while a distance-$r$ dominating set is a set $D\subseteq V(G)$ such that every vertex of the graph is within distance at most $r$ from a vertex from $D$. We study the duality between the maximum size of a distance-$2r$ independent set and the minimum size of a distance-$r$ dominating set in nowhere dense graph classes, as well as the kernelization complexity of the distance-$r$ independent set problem on these graph classes. Specifically, we prove that the distance-$r$ independent set problem admits an almost linear kernel on every nowhere dense graph class.

cs.DM↗
arXiv2020

Polynomial bounds for centered colorings on proper minor-closed graph classes

For $p\in \mathbb{N}$, a coloring $λ$ of the vertices of a graph $G$ is {\em{$p$-centered}} if for every connected subgraph~$H$ of $G$, either $H$ receives more than $p$ colors under $λ$ or there is a color that appears exactly once in $H$. In this paper, we prove that every $K_t$-minor-free graph admits a $p$-centered coloring with $\mathcal{O}(p^{g(t)})$ colors for some function $g$. In the special case that the graph is embeddable in a fixed surface $Σ$ we show that it admits a $p$-centered coloring with $\mathcal{O}(p^{19})$ colors, with the degree of the polynomial independent of the genus of $Σ$. This provides the first polynomial upper bounds on the number of colors needed in $p$-centered colorings of graphs drawn from proper minor-closed classes, which answers an open problem posed by Dvoř{á}k. As an algorithmic application, we use our main result to prove that if $\mathcal{C}$ is a fixed proper minor-closed class of graphs, then given graphs $H$ and $G$, on $p$ and $n$ vertices, respectively, where $G\in \mathcal{C}$, it can be decided whether $H$ is a subgraph of $G$ in time $2^{\mathcal{O}(p\log p)}\cdot n^{\mathcal{O}(1)}$ and space $n^{\mathcal{O}(1)}$.

cs.DM↗
arXiv2020

Vertex Cover Reconfiguration and Beyond

In the Vertex Cover Reconfiguration (VCR) problem, given a graph $G$, positive integers $k$ and $\ell$ and two vertex covers $S$ and $T$ of $G$ of size at most $k$, we determine whether $S$ can be transformed into $T$ by a sequence of at most $\ell$ vertex additions or removals such that every operation results in a vertex cover of size at most $k$. Motivated by results establishing the W[1]-hardness of VCR when parameterized by $\ell$, we delineate the complexity of the problem restricted to various graph classes. In particular, we show that VCR remains W[1]-hard on bipartite graphs, is NP-hard, but fixed-parameter tractable on (regular) graphs of bounded degree and more generally on nowhere dense graphs and is solvable in polynomial time on trees and (with some additional restrictions) on cactus graphs.

cs.CC↗
arXiv2020

Rankwidth meets stability

We study two notions of being well-structured for classes of graphs that are inspired by classic model theory. A class of graphs $C$ is monadically stable if it is impossible to define arbitrarily long linear orders in vertex-colored graphs from $C$ using a fixed first-order formula. Similarly, monadic dependence corresponds to the impossibility of defining all graphs in this way. Examples of monadically stable graph classes are nowhere dense classes, which provide a robust theory of sparsity. Examples of monadically dependent classes are classes of bounded rankwidth (or equivalently, bounded cliquewidth), which can be seen as a dense analog of classes of bounded treewidth. Thus, monadic stability and monadic dependence extend classical structural notions for graphs by viewing them in a wider, model-theoretical context. We explore this emerging theory by proving the following: - A class of graphs $C$ is a first-order transduction of a class with bounded treewidth if and only if $C$ has bounded rankwidth and a stable edge relation (i.e. graphs from $C$ exclude some half-graph as a semi-induced subgraph). - If a class of graphs $C$ is monadically dependent and not monadically stable, then $C$ has in fact an unstable edge relation. As a consequence, we show that classes with bounded rankwidth excluding some half-graph as a semi-induced subgraph are linearly $χ$-bounded. Our proofs are effective and lead to polynomial time algorithms.

cs.DM↗
arXiv2020

On the Generalised Colouring Numbers of Graphs that Exclude a Fixed Minor

The generalised colouring numbers $\mathrm{col}_r(G)$ and $\mathrm{wcol}_r(G)$ were introduced by Kierstead and Yang as a generalisation of the usual colouring number, and have since then found important theoretical and algorithmic applications. In this paper, we dramatically improve upon the known upper bounds for generalised colouring numbers for graphs excluding a fixed minor, from the exponential bounds of Grohe et al. to a linear bound for the $r$-colouring number $\mathrm{col}_r$ and a polynomial bound for the weak $r$-colouring number $\mathrm{wcol}_r$. In particular, we show that if $G$ excludes $K_t$ as a minor, for some fixed $t\ge4$, then $\mathrm{col}_r(G)\le\binom{t-1}{2}\,(2r+1)$ and $\mathrm{wcol}_r(G)\le\binom{r+t-2}{t-2}\cdot(t-3)(2r+1)\in\mathcal{O}(r^{\,t-1})$. In the case of graphs $G$ of bounded genus $g$, we improve the bounds to $\mathrm{col}_r(G)\le(2g+3)(2r+1)$ (and even $\mathrm{col}_r(G)\le5r+1$ if $g=0$, i.e. if $G$ is planar) and $\mathrm{wcol}_r(G)\le\Bigl(2g+\binom{r+2}{2}\Bigr)\,(2r+1)$.

math.CO↗
arXiv2020

Regular partitions of gentle graphs

Szemeredi's Regularity Lemma is a very useful tool of extremal combinatorics. Recently, several refinements of this seminal result were obtained for special, more structured classes of graphs. We survey these results in their rich combinatorial context. In particular, we stress the link to the theory of (structural) sparsity, which leads to alternative proofs, refinements and solutions of open problems. It is interesting to note that many of these classes present challenging problems. Nevertheless, from the point of view of regularity lemma type statements, they appear as "gentle" classes.

math.CO↗
arXiv2019

Linear rankwidth meets stability

Classes with bounded rankwidth are MSO-transductions of trees and classes with bounded linear rankwidth are MSO-transductions of paths. These results show 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: 1) Graphs with linear rankwidth at most $r$ are linearly \mbox{$χ$-bounded}. Actually, they have bounded $c$-chromatic number, meaning that they can be colored with $f(r)$ colors, each color inducing a cograph. 2) Based on a Ramsey-like argument, we prove for every proper hereditary family $\mathcal F$ 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~$\mathcal F$. 3) For a class $\mathcal C$ with bounded linear rankwidth the following conditions are equivalent: a) $\mathcal C$~is~stable, b)~$\mathcal C$~excludes some half-graph as a semi-induced subgraph, c) $\mathcal C$ 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 the Parameterized Complexity of Reconfiguration of Connected Dominating Sets

In a reconfiguration version of an optimization problem $\mathcal{Q}$ the input is an instance of $\mathcal{Q}$ and two feasible solutions $S$ and $T$. The objective is to determine whether there exists a step-by-step transformation between $S$ and $T$ such that all intermediate steps also constitute feasible solutions. In this work, we study the parameterized complexity of the \textsc{Connected Dominating Set Reconfiguration} problem (\textsc{CDS-R)}. It was shown in previous work that the \textsc{Dominating Set Reconfiguration} problem (\textsc{DS-R}) parameterized by $k$, the maximum allowed size of a dominating set in a reconfiguration sequence, is fixed-parameter tractable on all graphs that exclude a biclique $K_{d,d}$ as a subgraph, for some constant $d \geq 1$. We show that the additional connectivity constraint makes the problem much harder, namely, that \textsc{CDS-R} is \textsf{W}$[1]$-hard parameterized by $k+\ell$, the maximum allowed size of a dominating set plus the length of the reconfiguration sequence, already on $5$-degenerate graphs. On the positive side, we show that \textsc{CDS-R} parameterized by $k$ is fixed-parameter tractable, and in fact admits a polynomial kernel on planar graphs.

cs.DS↗