Abstract

Given a 2-coloring of the vertices of a regular n-gon P , how many parallel lines are needed to separate the vertices into monochromatic subsets? We prove that bn=2c is a tight upper bound, and also provide an O(n log n) time algorithm to determine the direction that gives the minimum number of lines. If the polygon is a non-regular convex polygon, then n 0 3 lines may be necessary, while n 0 2 lines always suffice. This problem arises in machine learning and has implications about the representational capabilities of some neural networks. 1 Introduction Let P be a regular polygon on n vertices. Given a 2-coloring of the vertices of P , how many parallel lines do we need to separate the vertices of P into monochromatic subsets? In other words, what is the smallest number m(n) of parallel lines that always suffices to partition the vertices of P into m(n) + 1 monochromatic subsets? Observe that an adversary picks the coloring of the vertices of P and we (the algorithm) choose the direc...

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