Dimension of CPT Posets

Abstract

A collection of linear orders on X, say ${\mathscr{L}}$, is said to realize a partially ordered set (or poset) $\mathcal {P} = (X, \preceq )$ if, for any two distinct x,y ∈ X, x ≼ y if and only if x ≺_Ly, $\forall L \in {\mathscr{L}}$. We call ${\mathscr{L}}$ a realizer of $\mathcal {P}$. The dimension of $\mathcal {P}$, denoted by $dim(\mathcal {P})$, is the minimum cardinality of a realizer of $\mathcal {P}$. A containment model $M_{\mathcal {P}}$ of a poset $\mathcal {P}=(X,\preceq )$ maps every x ∈ X to a set M_x such that, for every distinct x,y ∈ X, x ≼ y if and only if $M_{x} \varsubsetneq M_{y}$. We shall be using the collection (M_x)_x∈X to identify the containment model $M_{\mathcal {P}}$. A poset $\mathcal {P}=(X,\preceq )$ is a Containment order of Paths in a Tree (CPT poset), if it admits a containment model $M_{\mathcal {P}}=(P_{x})_{x \in X}$ where every P_x is a path of a tree T, which is called the host tree of the model. We show that if a poset $\mathcal {P}$ admits a CPT model in a host tree T of maximum degree Δ and radius r, then $dim(\mathcal {P}) \leq \lg \lg {\Delta } + (\frac {1}{2} + o(1))\lg \lg \lg {\Delta } + \lg r + \frac {1}{2} \lg \lg r + \frac {1}{2}\lg \pi + 3$. This bound is asymptotically tight up to an additive factor of $\min \limits (\frac {1}{2}\lg \lg \lg {\Delta }, \frac {1}{2}\lg \lg r)$. Further, let $\mathcal {P}(1,2;n)$ be the poset consisting of all the 1-element and 2-element subsets of [n] under ‘containment’ relation and let dim(1,2;n) denote its dimension. The proof of our main theorem gives a simple algorithm to construct a realizer for $\mathcal {P}(1,2;n)$ whose cardinality is only an additive factor of at most $\frac {3}{2}$ away from the optimum.