Helly-Type Theorems in Property Testing

Abstract

Helly’s theorem is a fundamental result in discrete geometry, describing the ways in which convex sets intersect with each other. If S is a set of n points in ℝ^d, we say that S is (k,G)-clusterable if it can be partitioned into k clusters (subsets) such that each cluster can be contained in a translated copy of a geometric object G. In this paper, as an application of Helly’s theorem, by taking a constant size sample from S, we present a testing algorithm for (k,G)-clustering, i.e., to distinguish between two cases: when S is (k,G)-clusterable, and when it is ε-far from being (k,G)-clusterable. A set S is ε-far (0 < ε ≤ 1) from being (k,G)-clusterable if at least εn points need to be removed from S to make it (k,G)-clusterable. We solve this problem for k = 1 and when G is a symmetric convex object. For k > 1, we solve a weaker version of this problem. Finally, as an application of our testing result, in clustering with outliers, we show that one can find the approximate clusters by querying a constant size sample, with high probability.