A Note on Constructive Canonical Splitter Strategies in Nowhere Dense Graph Classes

FPT algorithms for domination in sparse graphs and beyond

Nowhere dense classes of graphs were introduced by Nesetril and Ossona de Mendez as a model for "sparsity" in graphs. It turns out that nowhere dense classes of graphs can be characterised in many different ways and have been shown to be equivalent to other concepts studied in areas such as (finite) model theory. Therefore, the concept of nowhere density seems to capture a natural property of graph classes generalising for example classes of graphs which exclude a fixed minor, have bounded degree or bounded local tree-width. In this paper we give a self-contained introduction to the concept of nowhere dense classes of graphs focussing on the various ways in which they can be characterised. We also briefly sketch algorithmic applications these characterisations have found in the literature.

We prove that whenever G is a graph from a nowhere dense graph class 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,eps)|A|^(1+eps), where r is any positive integer, eps is any positive real, and f is a function that depends only on the class 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., CoRR, 2016]. As an algorithmic application of the above result, we show that for every fixed integer 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., STACS 2016], 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.

Nowhere dense graph classes, introduced by Nešetřil and Ossona de Mendez [2010, 2011], 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 (parameterized by the length of the input formula). 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 neighborhood covers for nowhere dense graphs. This extends and improves previous constructions of neighborhood 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 characterization 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.

Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez [30], form a large variety of classes of 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.

Kernelization and approximation of distance-[formula omitted] independent sets on nowhere dense graphs

Nowhere dense classes of graphs [21, 22] 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 article, we use tools from logic, in particular from a sub-field 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 radius r i N 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 C of graphs that is closed under taking subgraphs, the distance-r dominating set problem admits a kernel on C for every value of r if, and only if, it already admits a polynomial kernel for every value of r (under the standard assumption of parameterized complexity theory that FPT ≠ W[2]).

Nowhere dense classes of graphs [21, 22] 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 sub-field of model theory known as stability theory, to establish polynomial bounds for the equivalence of nowhere denseness and uniform quasi-wideness.As an algorithmic application of our new methods, we obtain for every fixed value of r ∊ ℕ a polynomial kernel for the distance-r dominating set problem on nowhere dense classes of graphs. This is particularly interesting, as it implies that for every subgraph-closed class C, the distance-r dominating set problem admits a kernel on 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 FPT ≠ W[2]).Finally, we demonstrate how to use the new methods to improve the parameter dependence of many fixed- parameter algorithms. As an example we provide a single exponential parameterized algorithm for the Connected Dominating Set problem on nowhere dense graph classes.

Nowhere dense classes of graphs [21, 22] 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 article, we use tools from logic, in particular from a sub-field 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 radius r ∈ N 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 C of graphs that is closed under taking subgraphs, the distance- r dominating set problem admits a kernel on C for every value of r if, and only if, it already admits a polynomial kernel for every value of r (under the standard assumption of parameterized complexity theory that FPT ≠ W[2]).

Over the past two decades the main focus of research into first-order (FO) model checking algorithms has been on sparse relational structures – culminating in the FPT algorithm by Grohe, Kreutzer and Siebertz for FO model checking on nowhere dense classes of graphs. On contrary to that, except the case of locally bounded clique-width only little is currently known about FO model checking on dense classes of graphs or other structures. We study the FO model checking problem on dense graph classes definable by geometric means (intersection and visibility graphs). We obtain new nontrivial FPT results, e.g., for restricted subclasses of circular-arc, circle, box, disk, and polygon-visibility graphs. These results use the FPT algorithm by Gajarský et al. for FO model checking on posets of bounded width. We also complement the tractability results by related hardness reductions.

On nowhere dense graphs

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2012 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Directed Nowhere Dense Classes of GraphsStephan Kreutzer and Siamak TazariStephan Kreutzer and Siamak Tazaripp.1552 - 1562Chapter DOI:https://doi.org/10.1137/1.9781611973099.123PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract Many natural computational problems on graphs such as finding dominating or independent sets of a certain size are well known to be intractable, both in the classical sense as well as in the framework of parameterized complexity. Much work therefore has focussed on exhibiting restricted classes of graphs on which these problems become tractable. While in the case of undirected graphs, there is a rich structure theory which can be used to develop tractable algorithms for these problems on large classes of undirected graphs, such a theory is much less developed for directed graphs. Many attempts to identify structure properties of directed graphs tailored towards algorithmic applications have focussed on a directed analogue of undirected tree-width. These attempts have proved to be successful in the development of algorithms for linkage problems but none of the existing width-measures allow for tractable solutions to important problems such as dominating sets and many other related problems. In this paper we take a radically different approach to identifying classes of directed graphs where domination and other problems become tractable. In particular, whereas most existing approaches treat the class of acyclic graphs as simple in their respective width measure, we will specifically study classes of digraphs which do not contain all acyclic digraphs. It is this new approach that make the algorithmic results reported herein possible. More specifically, we introduce the concept of shallow directed minors and based on this a new classification of classes of directed graphs which is diametric to existing directed graph decompositions and directed width measures proposed in the literature. We then study in depth one type of classes of directed graphs which we call nowhere crownful. The classes are very general as they include, on the one hand, all classes of directed graphs whose underlying undirected class is nowhere dense, such as planar, bounded-genus, and H-minor-free graphs; and on the other hand, also contain classes of high edge density whose underlying class is not nowhere dense. Yet we are able to show that problems such as directed dominating set and many others become fixed-parameter tractable on nowhere crownful classes of directed graphs. This is of particular interest as these problems are not tractable on any existing digraph measure for sparse classes. The algorithmic results are established via proving a structural equivalence of nowhere crownful classes and classes of graphs which are directed uniformly quasi-wide. While this result is inspired by [Nešetřil and Ossona de Mendez 2008], their proof method does not extend to the directed case and a different and much more involved proof is needed, turning it into a particularly significant part of our contribution. Previous chapter Next chapter RelatedDetails Published:2012ISBN:978-1-61197-210-8eISBN:978-1-61197-309-9 https://doi.org/10.1137/1.9781611973099Book Series Name:ProceedingsBook Code:PR141Book Pages:xiii + 1757

Meta-theorems for polynomial (linear) kernels have been the subject of intensive research in parameterized complexity. Heretofore, meta-theorems for linear kernels exist on graphs of bounded genus, $H$-minor-free graphs, and $H$-topological-minor-free graphs. To the best of our knowledge, no meta-theorems for polynomial kernels are known for any larger sparse graph classes; e.g., for classes of bounded expansion or for nowhere dense ones. In this paper we prove such meta-theorems for the two latter cases. More specifically, we show that graph problems that have finite integer index (FII) have linear kernels on graphs of bounded expansion when parameterized by the size of a modulator to constant-treedepth graphs. For nowhere dense graph classes, our result yields almost-linear kernels. While our parameter may seem rather strong, we argue that a linear kernelization result on graphs of bounded expansion with a weaker parameter (than treedepth modulator) would fail to include some of the problems covered by our framework. Moreover, we only require the problems to have FII on graphs of constant treedepth. This allows us to prove linear kernels for problems such as Longest Path/Cycle, Exact $s,t$-Path, Treewidth, and Pathwidth, which do not have FII on general graphs (and the first two not even on bounded treewidth graphs).

We prove that for every class of graphs ℒ which is nowhere dense, as defined by Nešetřil and Ossona de Mendez [28, 29], and for every first order formula φ(x, y), whenever one draws a graph G ∈ ℒ and a subset of its nodes A, the number of subsets of A|y| which are of the form {u ∈ A|y|: G |= φ(ū, v)} for some valuation ū of x in G is bounded by O(|A||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 [1] 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).

Bounded expansion and nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez, form a large variety of classes of uniformly sparse graphs which includes the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs. Since their initial definition it was shown that these graph classes can be defined in many equivalent ways: by generalised colouring numbers, neighbourhood complexity, sparse neighbourhood covers, a game known as the splitter game, and many more. We study the corresponding concepts for directed graphs. We show that the densities of bounded depth directed minors and bounded depth topological minors relate in a similar way as in the undirected case. We provide a characterisation of bounded expansion classes by a directed version of the generalised colouring numbers. As an application we show how to construct sparse directed neighbourhood covers and how to approximate directed distance-r dominating sets on classes of bounded expansion. On the other hand, we show that linear neighbourhood complexity does not characterise directed classes of bounded expansion.