Approximating the Colorful Caratheodory Theorem (システム数理と応用)
Let P1, . . . , Pd+1 ⊂ R be d-dimensional point sets such that the convex hull of each Pi contains the origin. We call the sets Pi color classes, and we think of the points in Pi as having color i. A colorful choice is a set with at most one point from each color class. The colorful Carathéodory theorem guarantees the existence of a colorful choice whose convex hull contains the origin. So far, the computational complexity of finding such a colorful choice is unknown. An m-colorful choice is a set that contains at most m points from each color class. We present an approximation algorithm that computes for any constant ε > 0, an dε(d + 1)e-colorful choice containing the origin in its convex hull in polynomial time. This notion of approximation has not been studied before, and it is motivated through the applications of the colorful Carathéodory theorem in the literature. Second, we show that the exact problem can be solved in d d) time if Θ(d log d) color classes are available, improving over the trivial d time algorithm.