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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 $existsmathbb{R}$ [ORourke: The complexity of computing minimum convex covers for polygons, Allerton 1982]. We prove that MCC is $existsmathbb{R}$-hard, and the problem is thus $existsmathbb{R}$-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 $existsmathbb{R}$-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 $mathsf{NP} eqexistsmathbb{R}$, 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.
We study several problems on geometric packing and covering with movement. Given a family $mathcal{I}$ of $n$ intervals of $kappa$ distinct lengths, and another interval $B$, can we pack the intervals in $mathcal{I}$ inside $B$ (respectively, cover $
Exactly 20 years ago at MFCS, Demaine posed the open problem whether the game of Dots & Boxes is PSPACE-complete. Dots & Boxes has been studied extensively, with for instance a chapter in Berlekamp et al. Winning Ways for Your Mathematical Plays, a w
Deciding whether a family of disjoint line segments in the plane can be linked into a simple polygon (or a simple polygonal chain) by adding segments between their endpoints is NP-hard.
Deciding whether a family of disjoint axis-parallel line segments in the plane can be linked into a simple polygon (or a simple polygonal chain) by adding segments between their endpoints is NP-hard.
Computing the similarity of two point sets is a ubiquitous task in medical imaging, geometric shape comparison, trajectory analysis, and many more settings. Arguably the most basic distance measure for this task is the Hausdorff distance, which assig