Line-Distortion, Bandwidth and Path-Length of a Graph

Abstract

For a graph $G=(V,E)$ the minimum line-distortion problem asks for the minimum k such that there is a mapping f of the vertices into points of the line such that for each pair of vertices x, y the distance on the line $|f(x) - f(y)|$ can be bounded by the term $d_G(x, y)\le |f(x)-f(y)|\le k \, d_G(x, y)$, where $d_G(x, y)$ is the distance in the graph. The minimum bandwidth problem minimizes the term $\max _{uv\in E}|f(u)-f(v)|$, where f is a mapping of the vertices of G into the integers $\{1, \ldots , n\}$. We investigate the minimum line-distortion and the minimum bandwidth problems on unweighted graphs and their relations with the minimum length of a Robertson–Seymour’s path-decomposition. The length of a path-decomposition of a graph is the largest diameter of a bag in the decomposition. The path-length of a graph is the minimum length over all its path-decompositions. In particular, we show:

• there is a simple polynomial time algorithm that embeds an arbitrary unweighted input graph G into the line with distortion $\mathcal{O}(k^2)$, where k is the minimum line-distortion of G;

• if a graph G can be embedded into the line with distortion k, then G admits a Robertson–Seymour’s path-decomposition with bags of diameter at most k in G;

• for every class of graphs with path-length bounded by a constant, there exist an efficient constant-factor approximation algorithm for the minimum line-distortion problem and an efficient constant-factor approximation algorithm for the minimum bandwidth problem;

• there is an efficient 2-approximation algorithm for computing the path-length of an arbitrary graph;

• AT-free graphs and some intersection families of graphs have path-length at most 2;

• for AT-free graphs, there exist a linear time 8-approximation algorithm for the minimum line-distortion problem and a linear time 4-approximation algorithm for the minimum bandwidth problem.