Minimum Linear Arrangement of Series-Parallel Graphs

TL;DR

This paper introduces an efficient approximation algorithm for the minimum linear arrangement problem specifically on series-parallel graphs, achieving a factor 14D^2 approximation with a simple divide-and-conquer approach.

Contribution

The paper presents the first study of the minimum linear arrangement problem on series-parallel graphs and provides a practical, parallelizable approximation algorithm with proven ratio.

Findings

Runs in linear time given a suitable decomposition

Achieves a 14D^2 approximation ratio

Effective for parallel implementation

Abstract

We present a factor $14D^2$ approximation algorithm for the minimum linear arrangement problem on series-parallel graphs, where $D$ is the maximum degree in the graph. Given a suitable decomposition of the graph, our algorithm runs in time $O(|E|)$ and is very easy to implement. Its divide-and-conquer approach allows for an effective parallelization. Note that a suitable decomposition can also be computed in time $O(|E|\log{|E|})$ (or even $O(\log{|E|}\log^*{|E|})$ on an EREW PRAM using $O(|E|)$ processors). For the proof of the approximation ratio, we use a sophisticated charging method that uses techniques similar to amortized analysis in advanced data structures. On general graphs, the minimum linear arrangement problem is known to be NP-hard. To the best of our knowledge, the minimum linear arrangement problem on series-parallel graphs has not been studied before.