Some distributions that allow perfect packing
Journal of the ACM (JACM)
Optimal bin packing with items of random sizes
Mathematics of Operations Research
Nonblocking multirate networks
SIAM Journal on Computing
On nonblocking multirate interconnection networks
SIAM Journal on Computing
Improving local search heuristics for some scheduling problems—I
Discrete Applied Mathematics - Special volume: first international colloquium on graphs and optimization (GOI), 1992
On Multirate Rearrangeable Clos Networks
SIAM Journal on Computing
On Rearrangeability of Multirate Clos Networks
SIAM Journal on Computing
ISCA '85 Proceedings of the 12th annual international symposium on Computer architecture
Multirate rearrangeable clos networks and a generalized edge coloring problem on bipartite graphs
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
The Differencing Method of Set Partitioning
The Differencing Method of Set Partitioning
Constructions of given-depth and optimal multirate rearrangeably nonblocking distributors
Journal of Combinatorial Optimization
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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. 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. Our analysis is interesting in 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.