A common thread in all the recent results concerning testing dense graphs is the use of Szemerédi's regularity lemma. In this paper we show that in some sense this is not a coincidence. Our first result is that the property defined by having any given Szemerédi-partition is testable with a constant number of queries. Our second and main result is a purely combinatorial characterization of the graph properties that are testable with a constant number of queries. This characterization (roughly) says that a graph property P can be tested with a constant number of queries if and only if testing P can be reduced to testing the property of satisfying one of finitely many Szemerédi-partitions. This means that in some sense, testing for Szemerédi-partitions is as hard as testing any testable graph property. We thus resolve one of the main open problems in the area of property-testing, which was first raised in the 1996 paper of Goldreich, Goldwasser and Ron [24] that initiated the study of graph property-testing. This characterization also gives an intuitive explanation as to what makes a graph property testable. Basic definitionsThe meta problem in the area of property testing is the following: Given a combinatorial structure S, distinguish between the case that S satisfies some property P and the case that S is -far from satisfying P. Roughly speaking, a combinatorial structure is said to be -far from satisfying some property P if an -fraction of its representation should be modified in order to make S satisfy P. The main goal is to design randomized algorithms, which look at a very small portion of the input, and using this information distinguish with high probability between the above two cases. Such algorithms are called property testers or simply testers for the property P. Preferably, a tester should look at a portion of the input whose size is a function of only. Blum, Luby and Rubinfeld [10] were the first to formulate a question of this type, and the general notion of property testing was first formulated by Rubinfeld and Sudan [34], who were interested in studying various algebraic properties such as linearity of functions.The main focus of this paper is the testing of properties of graphs. More specifically, we focus on property testing in the dense graph model as defined in [24]. In this case a graph G is said to be -far from satisfying a property P, if one needs to add/delete at least n 2 edges to G in order to turn it into a graph satisfying P. A tester for P should distinguish with high probability, say 2/3, between the case that G satisfies P and the case that G is -far from satisfying P. Here we assume that the tester can query some oracle whether a pair of vertices, i and j, are adjacent in the input graph G. In what follows we will say that a tester for a graph property P has one-sided error if it accepts any graph satisfying P with probability 1 (and still rejects those that are -far with probability at least 2/3). If the tester may reject graphs satisfying P with non-zero probability then...

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In this paper we study the Target Set Selection problem proposed by Kempe, Kleinberg, and Tardos; a problem which gives a nice clean combinatorial formulation for many applications arising in economy, sociology, and medicine. Its input is a graph with vertex thresholds, the social network, and the goal is to find a subset of vertices, the target set, that "activates" a prespecified number of vertices in the graph. Activation of a vertex is defined via a so-called activation process as follows: Initially, all vertices in the target set become active. Then at each step i of the process, each vertex gets activated if the number of its active neighbors at iteration i − 1 exceeds its threshold. The activation process is "monotone" in the sense that once a vertex is activated, it remains active for the entire process. Our contribution is as follows: First, we present an algorithm for Target Set Selection running in n O(w) time, for graphs with n vertices and treewidth bounded by w. This algorithm can be adopted to much more general settings, including the case of directed graphs, weighted edges, and weighted vertices. On the other hand, we also show that it is highly unlikely to find an n o(√ w) time algorithm for Target Set Selection, as this would imply a sub-exponential algorithm for all problems in SNP. Together with our upper bound result, this shows that the treewidth parameter determines the complexity of Target Set Selection to a large extent, and should be taken into consideration when tackling this problem in any scenario. In the last part of the paper we also deal with the "non-monotone" variant of Target Set Selection, and show that this problem becomes #P-hard on graphs with edge weights.