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.

中文翻译：

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覆盖多边形更难

在 MINIMUM CONVEX COVER (MCC) 问题中，我们给出了一个简单的多边形 $\mathcal P$ 和一个整数 $k$，问题是是否存在 $k$ 并集为 $\mathcal P$ 的凸多边形。众所周知，MCC 是 $\mathsf{NP}$-hard [Culberson & Reckhow：覆盖多边形很难，FOCS 1988/Journal of Algorithms 1994] 和 $\exists\mathbb{R}$ [O'Rourke: The计算多边形最小凸覆盖的复杂性，Allerton 1982]。我们证明 MCC 是 $\exists\mathbb{R}$-hard，因此问题是 $\exists\mathbb{R}$-complete。换句话说，这个问题等价于判断一个由整数系数的多项式方程和不等式组成的系统是否有实数解。如果我们构建的多边形的覆盖层存在，那么完全由三角形组成的覆盖层也存在。作为副产品，因此，我们还确定 $\exists\mathbb{R}$-complete 决定 $k$ 三角形是否覆盖给定的多边形。不知道是否在 $\mathsf{NP}$ 中找到最小覆盖的问题在文献中一再被提出，并且在 2001 年就被提及为一个“长期存在的悬而未决的问题”[Eidenbenz & Widmayer：具有对数性能保证的最小凸覆盖的近似算法，ESA 2001/SIAM Journal on Computing 2003]。我们证明，假设普遍认为 $\mathsf{NP}\neq\exists\mathbb{R}$，问题不在 $\mathsf{NP}$ 中。结果的一个含义是，在某些情况下，许多寻找小覆盖的自然方法必然会给出次优解决方案，