The multicast packing problem

This paper presents algorithms, heuristics and lower bounds for an optimal sharing of network resources among several multicast groups that coexist in the network. Group (i.e., many-to-many) multicasting is a demanding service since any member can become a sender independently from the others. We consider a shared tree as the backbone of a group multicasting session. Considering each multicast session in isolation and independently may cause congestion on some links and reduce network utilization. Thus, we define the multicast packing problem in which the network tries to accommodate simultaneously all the multicast groups while trying to avoid bottlenecks on the links for higher throughput (i.e., minimize the maximum link sharing among multicast groups). Minimization of maximum congestion is achieved at the expense of increasing the size of some multicast trees which in turn impacts the delay. This trade-off is addressed by adding a penalty term to the objective function of the optimal packing formulation. The penalty term is a function of the amount of dilation from the size of the optimal tree obtained for each group multicast independently from the others (i.e., in isolation). Since the mathematical programming formulation for the optimization problem is computationally intractable, we resort to suboptimal solutions with heuristics. Our heuristic method aims to reduce the sharing of a link while ensuring that the size of multicast trees will never exceed αOPT^k where OPT^k is the size of the optimum tree for multicast group k in isolation. Optimum multicast tree for each group (in isolation) is computed by using cutting-plane inequalities and the branch-and-cut algorithm. In order to evaluate the performance of our approximation, we derive lower bounds on the problem. Our first lower bound on the maximum congestion is a theoretical one and puts a cap on the following two constructive lower bounds. The lower bounds and the heuristic method are implemented and it is shown that the maximum congestion obtained by the heuristic method is quite close to the constructive lower bounds