On Embeddability of Buses in Point Sets

TL;DR

This paper investigates the bus embeddability problem, determining conditions for planar realizations of colored points with horizontal buses, and provides algorithms and complexity results for various problem variants.

Contribution

It introduces ILP and FPT algorithms for BEP, analyzes restricted cases with efficient solutions, and proves NP-completeness for a specific variant.

Findings

ILP and FPT algorithms for general BEP

Efficient algorithms for restricted BEP variants

NP-completeness of a special BEP case

Abstract

Set membership of points in the plane can be visualized by connecting corresponding points via graphical features, like paths, trees, polygons, ellipses. In this paper we study the \emph{bus embeddability problem} (BEP): given a set of colored points we ask whether there exists a planar realization with one horizontal straight-line segment per color, called bus, such that all points with the same color are connected with vertical line segments to their bus. We present an ILP and an FPT algorithm for the general problem. For restricted versions of this problem, such as when the relative order of buses is predefined, or when a bus must be placed above all its points, we provide efficient algorithms. We show that another restricted version of the problem can be solved using 2-stack pushall sorting. On the negative side we prove the NP-completeness of a special case of BEP.