In this paper, we consider the question of determining whether a function f has property P or is ⑀-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a -Clique (clique of density with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

Abstract.We propose and analyze a distribution learning algorithm for variable memory length Markov processes. These processes can be described by a subclass of probabilistic finite automata which we name Probabilistic Suffix Automata (PSA). Though hardness results are known for learning distributions generated by general probabilistic automata, we prove that the algorithm we present can efficiently learn distributions generated by PSAs. In particular, we show that for any target PSA, the KL-divergence between the distribution generated by the target and the distribution generated by the hypothesis the learning algorithm outputs, can be made small with high confidence in polynomial time and sample complexity. The learning algorithm is motivated by applications in human-machine interaction, Here we present two applications of the algorithm. In the first one we apply the algorithm in order to construct a model of the English language, and use this model to correct corrupted text. In the second application we construct a simple stochastic model for E.coli DNA.

We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Whereas they view graphs as represented by their adjacency matrix and measure distance between graphs ss a fraction of all possible vertex pairs, we view graphs as represented by boundedlength incidence lists and measure distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms for testing whether an unknown bounded-degree graph is connected, k-connected (for k > 1), planar, etc. Our algorithms work in time polynomial in 1/6, always accept the graph when it has the tested property, and reject with high probability if the graph is c-away from having the property. For example, the 2-Connectivity algorithm rejects (w. h.p. ) any .V-vertex d-degree graph for which more than c~(dJV) edges need to be added to make the graph 2-edge-connected.In addition we prove lower bounds of~(w) on the query complexity of testing algorithms for the Bipartite and Expander properties.

Abstract. We consider testing graph expansion in the bounded-degree graph model. Specifically, we refer to algorithms for testing whether the graph has a second eigenvalue bounded above by a given threshold or is far from any graph with such (or related) property. We present a natural algorithm aimed towards achieving the foregoing task. The algorithm is given a (normalized) eigenvalue bound λ < 1, oracle access to a bounded-degree N -vertex graph, and two additional parameters ǫ, α > 0. The algorithm runs in time N 0.5+α /poly(ǫ), and accepts any graph having (normalized) second eigenvalue at most λ. We believe that the algorithm rejects any graph that is ǫ-far from having second eigenvalue at most λ α/O(1) , and prove the validity of this belief under an appealing combinatorial conjecture.

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