Notes on Lower-Bounds for Testing
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چکیده
Recall that we showed that there is an algorithm for testing triangle-freeness of dense graphs that had a dependence on 1/ that was quite high. While we cannot exactly match that dependence with a lower bound, we’ll show that a super-polynomial dependence is necessary. Here we’ll give the proof only for the one-sided error case, and note that it can be extended to two-sided error algorithms. Namely, we’ll show how to construct dense graphs that are -far from being trianglefree but for which it is necessary to perform a super-polynomial number of queries in order to see a triangle. We’ll use a fact (without proving it), that in the case of dense graphs, if we are willing to accept a quadratic increase in the query complexity then we may assume without loss of generality that the algorithm takes a uniformly selected sample of vertices and makes its decision based on the induced subgraph. This “gets rid” of having to deal with adaptivity. Furthermore, since we are currently considering one-sided error algorithms, the algorithm may reject only if it obtains a triangle in the sample. The construction we’ll see is based on graphs that are known as Behrend Graphs. These graphs are defined by sets of integers that include no three-term arithmetic progression (abbreviated as 3AP). Namely, these are sets X ⊂ {1, . . . ,m} such that for every three elements x1, x2, x3 ∈ X, if x2 − x1 = x3 − x2 (i.e., x1 + x3 = 2x2), then necessarily x1 = x2 = x3. Below we describe a construction of such sets that are large (relative to m), and later explain how such sets determine Behrend graphs.
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تاریخ انتشار 2006