The Edge Group Coloring Problem with Applications to Multicast Switching

TL;DR

This paper studies a generalized edge coloring problem relevant to multicast switching, demonstrating its NP-hardness and evaluating approximation algorithms that perform well under certain conditions.

Contribution

It introduces a new generalized edge coloring problem, analyzes its computational complexity, and assesses approximation algorithms both theoretically and experimentally.

Findings

Best algorithms approximate the chromatic number within a small constant factor for random graphs.

When output/input ratio is less than 10, algorithms use fewer than 2χ colors.

One algorithm finds high-quality solutions with high probability for high-ratio graphs.

Abstract

This paper introduces a natural generalization of the classical edge coloring problem in graphs that provides a useful abstraction for two well-known problems in multicast switching. We show that the problem is NP-hard and evaluate the performance of several approximation algorithms, both analytically and experimentally. We find that for random $\chi$-colorable graphs, the number of colors used by the best algorithms falls within a small constant factor of $\chi$, where the constant factor is mainly a function of the ratio of the number of outputs to inputs. When this ratio is less than 10, the best algorithms produces solutions that use fewer than $2\chi$ colors. In addition, one of the algorithms studied finds high quality approximate solutions for any graph with high probability, where the probability of a low quality solution is a function only of the random choices made by the algorithm.