In this paper we present a dual approximation scheme for the class constrained shelf bin packing problem. In this problem, we are given bins of capacity 1, and n items of Q different classes, each item e with class ce and size se. The problem is to pack the items into bins, such that two items of different classes packed in a same bin must be in different shelves. Items in a same shelf are packed consecutively. Moreover, items in consecutive shelves must be separated by shelf divisors of size d. In a shelf bin packing problem, we have to obtain a shelf packing such that the total size of items and shelf divisors in any bin is at most 1. A dual approximation scheme must obtain a shelf packing of all items into N bins, such that, the total size of all items and shelf divisors packed in any bin is at most 1 + ε for a given ε > 0 and N is the number of bins used in an optimum shelf bin packing problem. Shelf divisors are used to avoid contact between items of different classes and can hold a set of items until a maximum given weight. We also present a dual approximation scheme for the class constrained bin packing problem. In this problem, there is no use of shelf divisors, but each bin uses at most C different classes.

The problem of minimizing the maximum edge congestion in a multicastcommunication network generalizes the well-known NP-hard multicommodityflow problem. We give the presently best theoretical approximation results aswell as efficient implementations. In particular we show that for a networkwith m edges and k multicast requests, anr(1 + ε)(rOPT + exp(1)lnm)-approximation can be computed inO(kmε-2 lnklnm) time, where β bounds the time forcomputing an r-approximate minimum Steiner tree. Moreover, we present a newfast heuristic that outperforms the primal-dual approaches with respect toboth running time and objective value.