Evolutionary Algorithms for One-Sided Bipartite Crossing Minimisation

TL;DR

One-Sided Bipartite Crossing Minimisation (OBCM) fixes the first-layer vertex order and seeks a permutation of the second layer that minimises edge crossings. The study evaluates three mutation operators—swap, exchange, and jump—within a $(1+1)$-type EA and randomised local search, compares them to deterministic baselines (Barycenter, Median, Nagamochi, Sifting), and solves the optimum on small instances with an ILP up to $n \approx 190$. Empirical results show that jump-based methods deliver the best solution quality given enough time, while jumps incur higher runtimes, mitigated by speed-up variants JFIRLS, JRIRLS, and JSRLS that can accelerate convergence by up to $100$ times with comparable final quality. Wilcoxon tests confirm the ordering swaps < exchange < jumps, and the work provides practical guidance on deploying EAs for OBCM in graph drawing.

Abstract

Evolutionary algorithms (EAs) are universal solvers inspired by principles of natural evolution. In many applications, EAs produce astonishingly good solutions. As they are able to deal with complex optimisation problems, they show great promise for hard problems encountered in the field of graph drawing.To complement recent theoretical advances in the analysis of EAs on graph drawing, we contribute a fundamental empirical study. We consider the so-called \textsc{One-Sided Bipartite Crossing Minimisation (OBCM)}: given two layers of a bipartite graph and a fixed horizontal order of vertices on the first layer, the task is to order the vertices on the second layer to minimise the number of edge crossings. We empirically analyse the performance of simple EAs for OBCM and compare different mutation operators on the underlying permutation ordering problem: exchanging two elements (\textit{exchange}), swapping adjacent elements (\textit{swap}) and jumping an element to a new position (\textit{jump}). EAs using jumps easily outperform all deterministic algorithms in terms of solution quality after a reasonable number of generations. We also design variations of the best-performing EAs to reduce the execution time for each generation. The improved EAs can obtain the same solution quality as before and run up to 100 times faster.

Figures (1)

• Figure 1: (a) Wall-clock times averaged over the same 100 random instances with increasing $n$. Evolutionary algorithms were stopped when no improvement was found throughout $n^{1.5}$ subsequent generations. The costs for initialising the crossing matrix for the EAs and the Sifting algorithm are subtracted and shown separately. (b) & (c) Difference between the final evolved crossing number (for EAs) or the returned crossing number (for deterministic algorithms) and the optimal crossing number plotted over generations for classical algorithms and evolutionary algorithms. The plots show averages taken over a suite of instances.