Navigating Posets with Few Maps

TL;DR

This paper introduces new parameters for finite posets related to their representation via plane maps, explores their theoretical bounds, and examines computational complexity aspects.

Contribution

It defines and analyzes mapability and atlas thickness parameters, relating them to classical poset measures, and studies their computational complexity.

Findings

Every 2-dimensional poset admits a map where all elements are tight.

The parameters relate to dimension and width: dim(P) ≤ 2*at(P) ≤ width(P)+1.

Computing mapability is NP-complete; fixed-parameter tractable with respect to certain parameters.

Abstract

We study two new parameters for finite posets motivated by the problem of efficiently determining the set of successors of a given element. A plane map of a poset $P=(X,\leq)$ is an injective mapping of $X$ into the Cartesian plane $\mathbb{R}^2$. Given two different points $a$ and $b$ in the plane, we say that $b$ dominates $a$ if $a<b$ coordinatewise. We say that an element $x$ of $P$ is tight in a plane map $\mu$ if the following holds: $x<y$ in $P$ if and only if $\mu(y)$ dominates $\mu(x)$. Note that, by definition, every 2-dimensional poset admits a map such that every element of the poset is tight. For any poset $P$, we define the mapability of $P$, $\mathrm{dmap}(P)$, to be the maximum number of elements that are tight in a single map, and we define the atlas thickness of $P$, $\mathrm{at}(P)$, to be the size of the smallest collection of maps such that every element is tight in at least one map of the collection. We relate these parameters to the classical notions of dimension and width: for every poset $P$, we show that $\mathrm{dim}(P) \le 2\mathrm{at}(P) \le \mathrm{width}(P)+1$. On the other hand, there exists a sequence of posets $(P_n)_{n \ge 1}$ such that the atlas thickness of $P_n$ is doubly exponential in the dimension of $P_n$. On the computational side, we prove that it is NP-complete, for a given poset $P$, to compute the mapability of $P$ and to decide whether $\mathrm{at}(P) \le 2$. In contrast to the latter, we show that computing the mapability of a poset is fixed-parameter tractable with respect to the natural parameter.