Combinatorial and Algorithmic Applications of the Borsuk-Ulam Theorem

The Borsuk-Ulam theorem states that if f : S → R is a continuous mapping from the unit sphere in R into R, there is a point x ∈ S where f(x) = f(−x); i.e., some pair of antipodal points has the same image. The recent book of Matoušek [21] is devoted to explaining this theorem, its background, and some of its many consequences in algebraic topology, algebraic geometry, and combinatorics. Borsuk-Ulam is a great theorem because it has several different equivalent versions, many different proofs, many extensions and generalizations, and many interesting applications. One familiar consequence is the ham-sandwich theorem (given d finite continuous measures on R, there exist a hyperplane that simultaneously bisects them), along with some of its extensions and generalizations to partitioning continuous measures [2], [6], [7], [8], [10]. In many cases there are discrete versions of these results, and it is interesting and instructive to find direct, combinatorial proofs. In addition, there are algorithmic issues about the computational complexity of finding the asserted combinatorial object. For example Lo et. al. [20] gave a direct combinatorial proof of the discrete ham-sandwich theorem and described algorithms to compute ham-sandwich cuts for point sets. Various generalizations and extensions were considered in [1], [2], [3], [4], [5], [9], [10], [11], [12], [17], [18], [19], [23], and [25]. A recent interesting example extends a result of Bárány and Matoušek [7], who combined Borsuk-Ulam with equivariant topology to show that three finite, continuous measures on R can be equipartitioned by a 2-fan, the partition of R determined by two half-lines incident at a point. Bereg [10] proved a discrete version of this result, and in a stronger form. He showed that given 2r red points, 2b blue points, and 2g green points in general position in R, and also given a line `, there is a point P ∈ ` and rays ρ1 and ρ2 incident at P (a 2-fan), that equipartition each set of points: i.e., r reds, b blues, g greens lie in the open wedge defined by the rays. He also described a beautiful algorithm to find such a partitioning. We show his algorithm is nearly optimal by proving the following result.