Covering Polygons is Even Harder

In the MINIMUM CONVEX COVER (MCC) problem, we are given a simple polygon $\mathcal P$ and an integer $k$, and the question is if there exist $k$ convex polygons whose union is $\mathcal P$. It is known that MCC is $\mathsf{NP}$-hard [Culberson & Reckhow: Covering polygons is hard, FOCS 1988/Journal of Algorithms 1994] and in $\exists\mathbb{R}$ [O'Rourke: The complexity of computing minimum convex covers for polygons, Allerton 1982]. We prove that MCC is $\exists\mathbb{R}$-hard, and the problem is thus $\exists\mathbb{R}$-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 $\exists\mathbb{R}$-complete to decide whether $k$ triangles cover a given polygon. The issue that it was not known if finding a minimum cover is in $\mathsf{NP}$ has repeatedly been raised in the literature, and it was mentioned as a "long-standing open question" already in 2001 [Eidenbenz & Widmayer: An approximation algorithm for minimum convex cover with logarithmic performance guarantee, ESA 2001/SIAM Journal on Computing 2003]. We prove that assuming the widespread belief that $\mathsf{NP}\neq\exists\mathbb{R}$, the problem is not in $\mathsf{NP}$. 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.