A linear-time algorithm for finding an edge-partition with max-min ratio at most two

Given a positive integer kk and an undirected edge-weighted connected simple graph GG with at least kk edges of positive weight, we wish to partition the graph into kk edge-disjoint connected components of approximately the same size. We focus on the max-min ratio of the partition, which is the weight of the maximum component divided by that of the minimum component. It has been shown that for some instances, the max-min ratio is at least two. In this paper, for any graph with no edge weight larger than one half of the average weight, we provide a linear-time algorithm for delivering a partition with max-min ratio at most two. Furthermore, by an extreme example, we show that the above restriction on edge weights is the loosest possible.