The Raymond and Beverly Sackler Faculty of Exact Sciences School of Computer Science Graph Property Testing and Related Problems

Property testers are fast randomized algorithms for distinguishing between objects satisfying a certain property from those that are 2-far from satisfying it. The focus of this thesis is in testing properties of graphs. This thesis is composed of three parts: In the first part of this thesis we study general testability results without much care how large the involved constants are as a function of the error parameter 2. In the first chapter we show that the entire family of hereditary properties can be tested with one-sided error. This result contains as a special case several previous results and also implies the testability of many well studied properties, which were not previously known to be testable. A few examples are the properties of being Perfect, Chordal, Interval and Ramsey. More interestingly, we use this result in order to give a characterization of the (natural) graph properties that can be tested with one sided error. One of the main open problems in the area of property testing, which was raised in the 1996 paper [75] of Goldreich, Goldwasser and Ron that initiated the study of graph property-testing, was to characterize the graph properties that can be tested with a constant number of queries. The second chapter resolves this open problem by giving a combinatorial characterization of testable properties. In the third chapter we study the relation between uniform and non-uniform property testers. We prove that there are (relatively) natural graph properties that can be tested by non-uniform testers but cannot be tested by uniform testers. In the second part we take a “closer” look at testing certain types of properties, and try to classify the properties that can be tested with a small number of queries. We first study properties defined by a forbidden induced graph H. In the second chapter (of the second part) we consider the property of not containing a copy of a given fixed directed graph D. We also consider the question of whether allowing two-sided error testers can improve the query complexity of testing the above problems. In both cases we give (nearly) complete characterizations of the graphs H (and digraphs D) for which the corresponding problems can be tested with a small number of queries. We also show that two-sided error testers cannot be efficient in the cases were there is no efficient one-sided error tester. The results of this part resolve several open problems that were raised by Alon [1]. In the third part we study algorithmic results that have some connection to the area of property testing. More specifically we study the following meta problem: given a graph G how well can we approximate the number of edges that need to be removed from G in order to make it satisfy a monotone graph property P. We first show that for any monotone property P and for every 2 > 0, this quantity can be approximated in linear time to within an additive error of 2n. A natural question is whether it is possible to obtain a better approximation in polynomial time. The second result gives a precise characterization of the monotone properties for which one can approximate the number of necessary edge deletions within an additive error of n2−δ. This characterization asserts that if there is a bipartite graph that does not satisfy P then such an algorithm is trivial, and in the other case the problem is NP -hard. This characterization resolves (in a strong form) a question raised by Yannakakis [115] in 1981.