# Random Structures & Algorithms

© Wiley Periodicals, Inc.

Edited By: Michal Karonski, Noga Alon, and Andrzej Rucinski

Impact Factor: 1.011

ISI Journal Citation Reports © Ranking: 2015: 52/106 (Computer Science Software Engineering); 57/312 (Mathematics); 89/254 (Mathematics Applied)

Online ISSN: 1098-2418

#### Virtual Issue - Property Testing

Property testing is an active recent area in Discrete Mathematics and Theoretical Computer Science, that deals with the problem of distinguishing between combinatorial structures that satisfy a certain property, and ones that are far from satisfying it.

This property testing virtual issue contains a selection of papers covering these key topics: testing of properties of graphs (dense graphs, bounded degree graphs, directed graphs), testing distributions, testing polynomials, languages and homomorphisms, and the connection to the theory of graph limits. The authors of the papers include many of the main contributors to the area.

This issue can thus be inspiring for anybody interested in the area, and can also be useful for those less familiar with it, providing a selection of papers that represent the challenges, methods and applications of the topic.

**Read all articles contained in this virtual issue for free...**

**Parameter testing in bounded degree graphs of subexponential growth**

*G. Elek*

*From the abstract:* Parameter testing algorithms are using constant number of queries to estimate the value of a certain parameter of a very large finite graph. It is well-known that graph parameters such as the independence ratio or the edit-distance from 3-colorability are not testable in bounded degree graphs. We prove, however, that these and several other interesting graph parameters are testable in bounded degree graphs of subexponential growth.

**On the testability and repair of hereditary hypergraph properties**

*T. Austin, T. Tao*

*From the abstract:* In this paper we make some refinements to these results of recent work of Alon–Shapira (A characterization of the (natural) graph properties testable with one-sided error) and Rödl–Schacht (Generalizations of the removal lemma) some of which may be surprising. In the positive direction, we strengthen the results to cover hereditary properties of multiple directed polychromatic graphs and hypergraphs….

**Testing monotone high-dimensional distributions**

*R. Rubinfeld, R. Servedio*

*From the abstract:* We study several natural problems of testing properties of monotone distributions over the n-dimensional Boolean cube, given access to random draws from the distribution being tested. We give a poly(n)-time algorithm for testing whether a monotone distribution is equivalent to or ϵ-far (in the L1 norm) from the uniform distribution…

**Testing low-degree polynomials over prime fields**

*C.S. Jutla, A.C. Patthal, A. Rudra, D. Zuckerman*

*From the abstract:* We present an efficient randomized algorithm to test if a given function f : Fpn -> Fp (where p is a prime) is a low-degree polynomial. This gives a local test for Generalized Reed-Muller codes over prime fields….

**Testing monotonicity over graph products**

*S. Halevy, E. Kushilevitz*

*From the abstract:* We consider the problem of monotonicity testing over graph products. Monotonicity testing is one of the central problems studied in the field of property testing. We present a testing approach that enables us to use known monotonicity testers for given graphs G1, G2, to test monotonicity over their product G1 × G2…..

**Non-abelian homomorphism testing, and distributions close to their self-convolutions**

*M. Ben-Or, D. Coppersmith, M. Luby, R. Rubinfeld*

*From the abstract:* In this paper, we study two questions related to the problem of testing whether a function is close to a homomorphism….

**Testing graphs for colorability properties**

*E. Fischer*

*From the abstract:* Let P be a property of graphs. An ϵ-test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than ϵ(2)n edges to make it satisfy P....

**Testing Subgraphs in Large Graphs**

*N. Alon*

*From the abstract:* Let H be a fixed graph with h vertices, let G be a graph on n vertices, and suppose that at least ϵn2 edges have to be deleted from it to make it H-free...

**Three Theorems Regarding Testing Graph Properties**

*O. Goldreich, L. Trevisan*

*From the abstract:* Property testing is a relaxation of decision problems in which it is required to distinguish YES-instances (i.e., objects having a predetermined property) from instances that are far from any YES-instance. We presents three theorems regarding testing graph properties in the adjacency matrix representation.

**Testing membership in parenthesis languages**

*M. Parnas, D. Ron, R. Rubinfeld*

*From the abstract:* We continue the investigation of properties defined by formal languages. This study was initiated by Alon et al. [1], who described an algorithm for testing properties defined by regular languages. Alon et al. also considered several context free languages, and in particular Dyck languages, which contain strings of properly balanced parentheses.

**Testing properties of directed graphs: acyclicity and connectivity**

*M. Bender, D. Ron*

*From the abstract:* This article initiates the study of testing properties of directed graphs. In particular, the article considers the most basic property of directed graphs—acyclicity. Because the choice of representation affects the choice of algorithm, the two main representations of graphs are studied.

**Testing the Diameter of Graphs**

*M. Parnas, D. Ron*

*From the abstract:* We propose a general model for testing graph properties, which extends and simplifies the bounded degree model of Goldreich and Ron [Property Testing in Bounded Degree Graphs, Proc. 31st Annual ACM Symposium on the Theory of Computing, 1997, pp. 406–415.]

**Left and right convergence of graphs with bounded degree**

*C. Borgs, J. Chayes, J. Kahn, L. Lovász*

*From the abstract:* The theory of convergent graph sequences has been worked out in two extreme cases, dense graphs and bounded degree graphs. One can define convergence in terms of counting homomorphisms from fixed graphs into members of the sequence (left-convergence), or counting homomorphisms into fixed graphs (right-convergence).