Finding hidden hubs and dominating sets in sparse graphs by randomized neighborhood queries

The concept of bounded expansion provides a robust way to capture sparse graph classes with interesting algorithmic properties. Most notably, every problem definable in first-order logic can be solved in linear time on bounded expansion graph classes. First-order interpretations and transductions of sparse graph classes lead to more general, dense graph classes that seem to inherit many of the nice algorithmic properties of their sparse counterparts.In this work we introduce lacon- and shrub-decompositions and use them to characterize transductions of bounded expansion graph classes and other graph classes. If one can efficiently compute sparse shrub- or lacon-decompositions of transductions of bounded expansion classes then one can solve every problem definable in first-order logic in linear time on these classes.

We prove that the distance-r dominating set, distance-r connected dominating set, distance-r vertex cover, and distance-r connected vertex cover problems admit constant factor approximations in the CONGEST model of distributed computing in a constant number of rounds on classes of sparse high-girth graphs. In this paper, sparse means bounded expansion, and high-girth means girth at least 4r+2. Our algorithm is quite simple; however, the proof of its approximation guarantee is non-trivial. To complement the algorithmic results, we show tightness of our approximation by providing a loosely matching lower bound on rings.Our result is the first to show the existence of constant-factor approximations in a constant number of rounds in non-trivial classes of graphs for distance-r covering problems.

We solve the Union-Find problem (UF) efficiently for the case the input is restricted to several graph classes, namely partial k-trees for any fixed k, d-dimensional grids for any fixed dimension d and for planar graphs. For the later we develop a technique of decomposing such a graph into small subgraphs, patching, that might be useful for other algorithmic problems on planar graphs, too.By efficiency we do not only mean “linear time” in a theoretical setting but also a practical reorganization of memory such that a dynamic data structures for UF is allocated consecutively and thus to reduce the amount of page fault produced by UF implementations drastically.

On Distance Matching Number in a Class of Sparse Random Graphs

We overview algorithmic results for classes of sparse graphs emphasizing new developments in this area. We focus on recently introduced classes of graphs with bounded expansion and nowhere-dense graphs and relate algorithmic meta-theorems for these classes of graphs to their analogues for proper minor-closed classes of graphs, classes of graphs with bounded tree-width, locally bounded tree-width and locally excluding a minor.

FPT algorithms for domination in sparse graphs and beyond

Attaining a detailed understanding of the various biological networks in an organism lies at the core of the emerging discipline of systems biology. A precise description of the relationships formed between genes, mRNA molecules, and proteins is a necessary step toward a complete description of the dynamic behavior of an organism at the cellular level, and toward intelligent, efficient, and directed modification of an organism. The importance of understanding such regulatory, signaling, and interaction networks has fueled the development of numerous in silico inference algorithms, as well as new experimental techniques and a growing collection of public databases. The Software Environment for BIological Network Inference (SEBINI) has been created to provide an interactive environment for the deployment, evaluation, and improvement of algorithms used to reconstruct the structure of biological regulatory and interaction networks. SEBINI can be used to analyze high-throughput gene expression, protein abundance, or protein activation data via a suite of state-of-the-art network inference algorithms. It also allows algorithm developers to compare and train network inference methods on artificial networks and simulated gene expression perturbation data. SEBINI can therefore be used by software developers wishing to evaluate, refine, or combine inference techniques, as well as by bioinformaticians analyzing experimental data. Networks inferred from the SEBINI software platform can be further analyzed using the Collective Analysis of Biological Interaction Networks (CABIN) tool, which is an exploratory data analysis software that enables integration and analysis of protein-protein interaction and gene-to-gene regulatory evidence obtained from multiple sources. The collection of edges in a public database, along with the confidence held in each edge (if available), can be fed into CABIN as one "evidence network," using the Cytoscape SIF file format. Using CABIN, one may increase the confidence in individual edges in a network inferred by an algorithm in SEBINI, as well as extend such a network by combining it with species-specific or generic information, e.g., known protein-protein interactions or target genes identified for known transcription factors. Thus, the combined SEBINI-CABIN toolkit aids in the more accurate reconstruction of biological networks, with less effort, in less time.A demonstration web site for SEBINI can be accessed from https://www.emsl.pnl.gov/SEBINI/RootServlet . Source code and PostgreSQL database schema are available under open source license. ronald.taylor@pnl.gov. For commercial use, some algorithms included in SEBINI require licensing from the original developers. CABIN can be downloaded from http://www.sysbio.org/dataresources/cabin.stm . mudita.singhal@pnl.gov.

We describe algorithms, based on Avis and Fukuda's reverse search paradigm, for listing all maximal independent sets in a sparse graph in polynomial time and delay per output. For bounded degree graphs, our algorithms take constant time per set generated; for minor-closed graph families, the time is O ( n ) per set, and for more general sparse graph families we achieve subquadratic time per set. We also describe new data structures for maintaining a dynamic vertex set S in a sparse or minor-closed graph family, and querying the number of vertices not dominated by S ; for minor-closed graph families the time per update is constant, while it is sublinear for any sparse graph family. We can also maintain a dynamic vertex set in an arbitrary m -edge graph and test the independence of the maintained set in time O (√m) per update. We use the domination data structures as part of our enumeration algorithms.

In recent years, a variety of randomized constructions of sketching matrices have been devised, that have been used in fast algorithms for numerical linear algebra problems, such as least squares regression, low-rank approximation, and the approximation of leverage scores. A key property of sketching matrices is that of subspace embedding. In this paper, we study sketching matrices that are obtained from bipartite graphs that are sparse, i.e., have left degree s that is small. In particular, we explore two popular classes of sparse graphs, namely, expander graphs and magical graphs. For a given subspace ${\mathcal{U}} \subseteq {{\mathbb{R}}^n}$ of dimension k, we show that the magical graph with left degree s = 2 yields a (1 ± ϵ) ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> -subspace embedding for ${\mathcal{U}}$, if the number of right vertices (the sketch size) $m = {\mathcal{O}}\left( {{k^2}/{\varepsilon ^2}} \right)$. The expander graph with $s = {\mathcal{O}}(\log k/\varepsilon )$ yields a subspace embedding for $m = {\mathcal{O}}\left( {k\log k/{\varepsilon ^2}} \right)$. We also discuss the construction of sparse sketching matrices with reduced randomness using expanders based on error-correcting codes. Empirical results on various synthetic and real datasets show that these sparse graph sketching matrices work very well in practice.

A graph is a mathematical structure consisting of a set of objects, which we call vertices, and links between pairs of objects, which we call edges. Graphs are used to model many problems arising in areas such as physics, sociology, and computer science. It is partially because of the simplicity of the definition of a graph that the concept can be so widely used. Nevertheless, when applied to a particular task, it is not always necessary to study graphs in all their generality, and it can be convenient to studying them from a restricted point of view. Restriction can come from requiring graphs to be embeddable in a particular surface, to admit certain types of decompositions, or by forbidding some substructure. A collection of graphs satisfying a fixed restriction forms a class of graphs. Many important classes of graphs satisfy that graphs belonging to it cannot have many edges in comparison with the number of vertices. Such is the case of classes with an upper bound on the maximum degree, and of classes excluding a fixed minor. Recently, the notion of classes with bounded expansion was introduced by Neˇsetˇril and Ossona de Mendez [62], as a generalisation of many important types of sparse classes. In this thesis we study chromatic and structural properties of classes with bounded expansion. We say a graph is k-degenerate if each of its subgraphs has a vertex of degree at most k. The degeneracy is thus a measure of the density of a graph. This notion has been generalised with the introduction, by Kierstead and Yang [47], of the generalised colouring numbers. These parameters have found applications in many areas of Graph Theory, including a characterisation of classes with bounded expansion. One of the main results of this thesis is a series of upper bounds on the generalised colouring numbers, for different sparse classes of graphs, such as classes excluding a fixed complete minor, classes with bounded genus and classes with bounded tree-width. We also study the following problem: for a fixed positive integer p, how many colours do we need to colour a given graph in such a way that vertices at distance exactly p get different colours? When considering classes with bounded expansion, we improve dramatically on the previously known upper bounds for the number of colours needed. Finally, we introduce a notion of addition of graph classes, and show various cases in which sparse classes can be summed so as to obtain another sparse class.

Let $G=(V,E)$ be an undirected graph on $n$ vertices and $\lambda:E\to 2^{\mathbb{N}}$ a mapping that assigns to every edge a non-empty set of positive integer labels. These labels can be seen as discrete times when the edge is present. Such a labeled graph ${\cal G}=(G,\lambda)$ is said to be temporally connected if a path exists with non-decreasing times from every vertex to every other vertex. In a seminal paper, Kempe, Kleinberg, and Kumar (STOC 2000) asked whether, given such a temporal graph, a sparse subset of edges can always be found whose labels suffice to preserve temporal connectivity---a temporal spanner. Axiotis and Fotakis (ICALP 2016) answered negatively by exhibiting a family of $\Theta(n^2)$-dense temporal graphs which admit no temporal spanner of density $o(n^2)$. The natural question is then whether sparse temporal spanners can always be found in some classes of dense graphs. In this paper, we answer this question affirmatively, by showing that if the underlying graph $G$ is a complete graph, then one can always find temporal spanners of density $O(n \log n)$. The best known result for complete graphs so far was that spanners of density ${n \choose 2}- \lfloor n/4 \rfloor = O(n^2)$ always exist. Our result is the first positive answer as to the existence of $o(n^2)$ sparse spanners in adversarial instances of temporal graphs since the original question by Kempe et al., focusing here on complete graphs. The proofs are constructive and directly adaptable as an algorithm.

The dynamics of social relations and the possibility of reaching the state of structural balance (Heider balance) under the influence of the temperature modeling the social noise level are discussed for interacting actors occupying nodes of classical random graphs. Depending on the graph density D, either a smooth crossover or a first-order phase transition from a balanced to an imbalanced state of the system is observed with an increase in the thermal noise level. The minimal graph density Dmin for which the first-order phase transition can be observed decreases with the system size N as Dmin∝N−0.58(1). For graph densities D&amp;gt;Dmin, the reduced critical temperature Tc⋆=Tc/Tc(D=1) increases with the graph density as Tc⋆∝D1.719(6) independently of the system size N.

Listing, generating or enumerating objects of specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. We study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality. We establish enumeration algorithms as well as lower and upper bounds for the maximum number of minimal connected dominating sets in such graphs. In particular, we present algorithms to enumerate all minimal connected dominating sets of chordal graphs in time O(1.7159^n), of split graphs in time O(1.3803^n), and of AT-free, strongly chordal, and distance-hereditary graphs in time O^*(3^{n/3}), where n is the number of vertices of the input graph. Our algorithms imply corresponding upper bounds for the number of minimal connected dominating sets for these graph classes.

Structure in sparse k-critical graphs

Even though a large number of I/O-efficient graph algorithms have been developed, a number of fundamental problems still remain open. For example, no space- and I/O-efficient algorithms are known for depth-first search or breadth-first search in sparse graphs. In this paper we present two new results on I/O-efficient depth-first search in an important class of sparse graphs, namely undirected embedded planar graphs. We develop a new efficient depth-first search algorithm and show how planar depth-first search in general can be reduced to planar breadth-first search. As part of the first result we develop the first I/O-efficient algorithm for finding a simple cycle separator of a biconnected planar graph. Together with other recent reducibility results, the second result provides further evidence that external memory breadth-first search is among the hardest problems on planar graphs.