Convex Quadrangulations of Bichromatic Point Sets

Abstract

We consider quadrangulations of red and blue points in the plane where each face is convex and no edge connects two points of the same color. In particular, we show that the following problem is NP-hard: Given a finite set [Formula: see text] of points with each point either red or blue, does there exist a convex quadrangulation of [Formula: see text] in such a way that the predefined colors give a valid vertex 2-coloring of the quadrangulation? We consider this as a step towards solving the corresponding long-standing open problem on monochromatic point sets.

Counting Polygon Triangulations is Hard

“…In computational geometry, it has been shown that counting the vertices or facets of highdimensional convex polytopes is #P-complete [28], and that computing the expected total length of the minimum spanning tree of a stochastic subset of three-dimensional points is #Phard [25]. Additionally, when testing existence of a geometric structure is hard [5,9,31,29,26] it is just hard to determine whether its count is nonzero. However, we know of no past hardness proofs for counting easy-to-construct two-dimensional structures.…”

Counting Polygon Triangulations is Hard

“…In computational geometry, it has been shown that counting the vertices or facets of highdimensional convex polytopes is #P-complete [27], and that computing the expected total length of the minimum spanning tree of a stochastic subset of three-dimensional points is #Phard [24]. Additionally, when testing existence of a geometric structure is hard [5,9,25,28,30] it is just hard to determine whether its count is nonzero. However, we know of no past hardness proofs for counting easy-to-construct two-dimensional structures.…”