Trimming of Graphs, with Application to Point Labeling

TL;DR

This paper introduces a graph trimming technique for bounded domino treewidth graphs and applies it to develop a polynomial-time approximation scheme for the map labeling problem, solving a major open question.

Contribution

It establishes that graphs with bounded domino treewidth are trimmable and uses this to create an efficient approximation scheme for point labeling.

Findings

Graphs of bounded domino treewidth are trimmable.

Polynomial-time approximation scheme for weighted point labeling.

Solves a major open problem in map labeling theory.

Abstract

For $t,g>0$, a vertex-weighted graph of total weight $W$ is $(t,g)$-trimmable if it contains a vertex-induced subgraph of total weight at least $(1-1/t)W$ and with no simple path of more than $g$ edges. A family of graphs is trimmable if for each constant $t>0$, there is a constant $g=g(t)$ such that every vertex-weighted graph in the family is $(t,g)$-trimmable. We show that every family of graphs of bounded domino treewidth is trimmable. This implies that every family of graphs of bounded degree is trimmable if the graphs in the family have bounded treewidth or are planar. Based on this result, we derive a polynomial-time approximation scheme for the problem of labeling weighted points with nonoverlapping sliding labels of unit height and given lengths so as to maximize the total weight of the labeled points. This settles one of the last major open questions in the theory of map labeling.