Separability, Boxicity, and Partial Orders

Abstract

AbstractA collection $$S=\{S_i, \ldots , S_n\}$$ S = { S i , … , S n } of disjoint closed convex sets in $$\mathbb {R}^d$$ R d is separable if there exists a direction (a non-zero vector) $$ \overrightarrow{v}$$ v → of $$\mathbb {R}^d$$ R d such that the elements of S can be removed, one at a time, by translating them an arbitrarily large distance in the direction $$ \overrightarrow{v}$$ v → without hitting another element of S. We say that $$S_i \prec S_j$$ S i ≺ S j if $$S_j$$ S j has to be removed before we can remove $$S_i$$ S i . The relation $$\prec $$ ≺ defines a partial order $$P(S,\prec )$$ P ( S , ≺ ) on S which we call the separability order of S and $$ \overrightarrow{v}$$ v → . A partial order $$P(X, \prec ')$$ P ( X , ≺ ′ ) on $$X=\{x_1, \ldots , x_n\}$$ X = { x 1 , … , x n } is called a separability order if there is a collection of convex sets S and a vector $$ \overrightarrow{v}$$ v → in some $$\mathbb {R}^d$$ R d such that $$x_i \prec ' x_j$$ x i ≺ ′ x j in $$P(X, \prec ')$$ P ( X , ≺ ′ ) if and only if $$S_i \prec S_j$$ S i ≺ S j in $$P(S,\prec )$$ P ( S , ≺ ) . We prove that every partial order is the separability order of a collection of convex sets in $$\mathbb {R}^4$$ R 4 , and that any poset of dimension 2 is the separability order of a set of line segments in $$\mathbb {R}^3$$ R 3 . We then study the case when the convex sets are restricted to be boxes in d-dimensional spaces. We prove that any partial order is the separability order of a family of disjoint boxes in $$\mathbb {R}^d$$ R d for some $$d \le \lfloor \frac{n}{2} \rfloor +1$$ d ≤ ⌊ n 2 ⌋ + 1 . We prove that every poset of dimension 3 has a subdivision that is the separability order of boxes in $$\mathbb {R}^3$$ R 3 , that there are partial orders of dimension 2 that cannot be realized as box separability in $$\mathbb {R}^3$$ R 3 and that for any d there are posets with dimension d that are separability orders of boxes in $$\mathbb {R}^3$$ R 3 . We also prove that for any d there are partial orders with box separability dimension d; that is, d is the smallest dimension for which they are separable orders of sets of boxes in $$\mathbb {R}^d$$ R d .