Every Minor-Closed Property of Sparse Graphs is Testable

Suppose $G$ is a graph with degrees bounded by $d$, and one needs to remove more than $\epsilon n$ of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of $G$ is far from the statistics of local neighborhoods around vertices of any planar graph $G'$ with the same degree bound. In fact, a similar result is proved for any minor-closed property of bounded degree graphs. As an immediate corollary of the above result we infer that many well studied graph properties, like being planar, outer-planar, series-parallel, bounded genus, bounded tree-width and several others, are testable with a constant number of queries, where the constant may depend on $\epsilon$ and $d$, but not on the graph size. None of these properties was previously known to be testable even with $o(n)$ queries

Every property is testable on a natural class of scale-free multigraphs

In this paper, we introduce a natural class of multigraphs called hierarchical-scale-free (HSF) multigraphs, and consider constant-time testability on the class. We show that a very wide subclass, specifically, that in which the power-law exponent is greater than two, of HSF is hyperfinite. Based on this result, an algorithm for a deterministic partitioning oracle can be constructed. We conclude by showing that every property is constant-time testable on the above subclass of HSF. This algorithm utilizes findings by Newman and Sohler of STOC'11. However, their algorithm is based on the bounded-degree model, while it is known that actual scale-free networks usually include hubs, which have a very large degree. HSF is based on scale-free properties and includes such hubs. This is the first universal result of constant-time testability on the general graph model, and it has the potential to be applicable on a very wide range of scale-free networks.Comment: 13 pages, one figure. Difference from ver. 1: Definitions of HSF and SF become more general. Typos were fixe

Approximating the Spectrum of a Graph

The spectrum of a network or graph $G=(V,E)$ with adjacency matrix $A$, consists of the eigenvalues of the normalized Laplacian $L= I - D^{-1/2} A D^{-1/2}$. This set of eigenvalues encapsulates many aspects of the structure of the graph, including the extent to which the graph posses community structures at multiple scales. We study the problem of approximating the spectrum $\lambda = (\lambda_1,\dots,\lambda_{|V|})$, $0 \le \lambda_1,\le \dots, \le \lambda_{|V|}\le 2$ of $G$ in the regime where the graph is too large to explicitly calculate the spectrum. We present a sublinear time algorithm that, given the ability to query a random node in the graph and select a random neighbor of a given node, computes a succinct representation of an approximation $\widetilde \lambda = (\widetilde \lambda_1,\dots,\widetilde \lambda_{|V|})$, $0 \le \widetilde \lambda_1,\le \dots, \le \widetilde \lambda_{|V|}\le 2$ such that $\|\widetilde \lambda - \lambda\|_1 \le \epsilon |V|$. Our algorithm has query complexity and running time $exp(O(1/\epsilon))$, independent of the size of the graph, $|V|$. We demonstrate the practical viability of our algorithm on 15 different real-world graphs from the Stanford Large Network Dataset Collection, including social networks, academic collaboration graphs, and road networks. For the smallest of these graphs, we are able to validate the accuracy of our algorithm by explicitly calculating the true spectrum; for the larger graphs, such a calculation is computationally prohibitive. In addition we study the implications of our algorithm to property testing in the bounded degree graph model

Testing bounded arboricity

In this paper we consider the problem of testing whether a graph has bounded arboricity. The family of graphs with bounded arboricity includes, among others, bounded-degree graphs, all minor-closed graph classes (e.g. planar graphs, graphs with bounded treewidth) and randomly generated preferential attachment graphs. Graphs with bounded arboricity have been studied extensively in the past, in particular since for many problems they allow for much more efficient algorithms and/or better approximation ratios. We present a tolerant tester in the sparse-graphs model. The sparse-graphs model allows access to degree queries and neighbor queries, and the distance is defined with respect to the actual number of edges. More specifically, our algorithm distinguishes between graphs that are $\epsilon$-close to having arboricity $\alpha$ and graphs that $c \cdot \epsilon$-far from having arboricity $3\alpha$, where $c$ is an absolute small constant. The query complexity and running time of the algorithm are $\tilde{O}\left(\frac{n}{\sqrt{m}}\cdot \frac{\log(1/\epsilon)}{\epsilon} + \frac{n\cdot \alpha}{m} \cdot \left(\frac{1}{\epsilon}\right)^{O(\log(1/\epsilon))}\right)$ where $n$ denotes the number of vertices and $m$ denotes the number of edges. In terms of the dependence on $n$ and $m$ this bound is optimal up to poly-logarithmic factors since $\Omega(n/\sqrt{m})$ queries are necessary (and $\alpha = O(\sqrt{m}))$. We leave it as an open question whether the dependence on $1/\epsilon$ can be improved from quasi-polynomial to polynomial. Our techniques include an efficient local simulation for approximating the outcome of a global (almost) forest-decomposition algorithm as well as a tailored procedure of edge sampling

A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error

We consider one-sided error property testing of $\mathcal{F}$-minor freeness in bounded-degree graphs for any finite family of graphs $\mathcal{F}$ that contains a minor of $K_{2,k}$, the $k$-circus graph, or the $(k\times 2)$-grid for any $k\in\mathbb{N}$. This includes, for instance, testing whether a graph is outerplanar or a cactus graph. The query complexity of our algorithm in terms of the number of vertices in the graph, $n$, is $\tilde{O}(n^{2/3} / \epsilon^5)$. Czumaj et~al.\ showed that cycle-freeness and $C_k$-minor freeness can be tested with query complexity $\tilde{O}(\sqrt{n})$ by using random walks, and that testing $H$-minor freeness for any $H$ that contains a cycles requires $\Omega(\sqrt{n})$ queries. In contrast to these results, we analyze the structure of the graph and show that either we can find a subgraph of sublinear size that includes the forbidden minor $H$, or we can find a pair of disjoint subsets of vertices whose edge-cut is large, which induces an $H$-minor.Comment: extended to testing outerplanarity, full version of ICALP pape

Random local algorithms

Consider the problem when we want to construct some structure on a bounded degree graph, e.g. an almost maximum matching, and we want to decide about each edge depending only on its constant radius neighbourhood. We show that the information about the local statistics of the graph does not help here. Namely, if there exists a random local algorithm which can use any local statistics about the graph, and produces an almost optimal structure, then the same can be achieved by a random local algorithm using no statistics.Comment: 9 page

Graph removal lemmas

The graph removal lemma states that any graph on n vertices with o(n^{v(H)}) copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and computer science. In this survey we discuss these lemmas, focusing in particular on recent improvements to their quantitative aspects.Comment: 35 page