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# Covering Polygons is Even Harder

### Abstract:

In the MINIMUM CONVEX COVER (MCC) problem, we are given a simple polygon P and an integer k, and the question is if there exist k convex polygons whose union is P. It is known that MCC is 𝖭𝖯-hard and in ∃ℝ. We prove that MCC is ∃ℝ-hard, and the problem is thus ∃ℝ-complete. In other words, the problem is equivalent to deciding whether a system of polynomial equations and inequalities with integer coefficients has a real solution.
If a cover for our constructed polygon exists, then so does a cover consisting entirely of triangles. As a byproduct, we therefore also establish that it is ∃ℝ-complete to decide whether k triangles cover a given polygon.
The issue that it was not known if finding a minimum cover is in 𝖭𝖯 has repeatedly been raised in the literature, and it was mentioned as a "long-standing open question" already in 2001. We prove that assuming the widespread belief that 𝖭𝖯≠∃ℝ, the problem is not in 𝖭𝖯.
An implication of the result is that many natural approaches to finding small covers are bound to give suboptimal solutions in some cases, since irrational coordinates of arbitrarily high algebraic degree can be needed for the corners of the pieces in an optimal solution.

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