Improved Bounds on Nonblocking 3-Stage Clos Networks

Abstract

We consider a generalization of edge coloring bipartite graphs in which every edge has a weight in $[0,1]$ and the coloring of the edges must satisfy that the sum of the weights of the edges incident to a vertex v of any color must be at most 1. For unit weights, König's theorem says that the number of colors needed is exactly the maximum degree. For this generalization, we show that $2.557 n + o(n)$ colors are sufficient, where n is the maximum total weight adjacent to any vertex, improving the previously best bound of $2.833n+O(1)$ due to Du et al. Our analysis is interesting on its own and involves a novel decomposition result for bipartite graphs and the introduction of an associated continuous one-dimensional bin packing instance which we can prove allows perfect packing. This question is motivated by the question of the rearrangeability of 3-stage Clos networks. In that context, the corresponding parameter n of interest in the edge coloring problem is the maximum over all vertices of the number of unit-sized bins needed to pack the weights of the incident edges. In that setting, we are able to improve the bound to $2.5480 n + o(n)$, also improving a bound of $2.5625n+O(1)$ of Du et al. We also consider the online version of this problem in which edges have to be colored as soon as they are revealed. In this context, we can show that $5n$ colors are enough. This contrasts with the best known lower bound of $3n-2$ by Tsai, Wang, and Hwang but improves upon the previous best upper bound of $5.75n$ obtained by Gao and Hwang. Additionally, we show several improved bounds for more restricted versions of the problem. These online bounds are achieved by simple and easy-to-implement algorithms, inspired by the first fit heuristic for bin packing.