Halving by a Thousand Cuts or Punctures
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For point sets P_1, …, P_, a set of lines L is halving if any face of the arrangement L contains at most |P_i|/2 points of P_i, for all i. We study the problem of computing a halving set of lines of minimal size. Surprisingly, we show a polynomial time algorithm that outputs a halving set of size O(^3/2), where is the size of the optimal solution. Our solution relies on solving a new variant of the weak -net problem for corridors, which we believe to be of independent interest. We also study other variants of this problem, including an alternative setting, where one needs to introduce a set of guards (i.e., points), such that no convex set avoiding the guards contains more than half the points of each point set.
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