On Dominance-Free Samples of a (Colored) Stochastic Dataset
  A point $p \in \mathbb{R}^d$ is said to dominate another point $q \in \mathbb{R}^d$ if the coordinate of $p$ is greater than or equal to the coordinate of $q$ in every dimension. A set of points in $\mathbb{R}^d$ is dominance-free if any two points do not dominate each other. We consider the problem of counting the dominance-free subsets of a given dataset in $\mathbb{R}^d$, or more generally, computing the probability that a random sample of a stochastic dataset in $\mathbb{R}^d$ where each point is sampled independently with its existence probability is dominance-free. In fact, we investigate a colored generalization of the problem, in which the points in the given stochastic dataset are colored and we are interested in the random samples that are inter-color dominance-free (i.e., any two points with different colors do not dominate each other). We propose the first algorithm that solves the problem for $d=2$ in near-quadratic time. On the other hand, we show that the problem is #P-hard for any $d \geq 3$, even if the points have a restricted color pattern; this implies the #P-hardness of the uncolored version (i.e., computing the dominance-free probability of a uncolored stochastic dataset) for $d \geq 3$. In addition, we show that even when the existence probabilities of the points are all equal to 0.5, the problem remains #P-hard for any $d \geq 7$; this implies the #P-hardness of counting dominance-free subsets for $d \geq 7$. In order to prove our hardness results, we establish some results about embedding the vertices a graph into low-dimensional Euclidean space such that two vertices are connected by an edge in the graph iff they form a dominance pair in the embedding. These results may be of independent interest and can possibly be applied to other problems. 
Jie Xue, Yuan Li
cs.CG