Pricing Multicasting in More Practical Network Models TITLE2:

• Authors: M. Adler;D. Rubenstein
• Venue: Pricing Multicasting in More Practical Network Models TITLE2:
• Year: 2001

In [9], Feigenbaum, Papadimitriou and Shenker initiate the study of pricing algorithms for multicast transmission. In this paper, we build on this work by studying the effect on the complexity of pricing when two practical considerations are incorporated into the network model. In particular, we study a model where the session is offered at a number of different rates of transmission, and where there is a cost for enabling multicasting at each node of the network. As a test case for the different types of pricing that can occur, we consider a pricing mechanism, called Marginal Cost, that has seen considerable attention in simpler network models. We demonstrate that the details of how multiple rates are provided has a significant impact on the complexity of pricing. For multiple rates provided via the layered paradigm, we provide a distributed algorithm for computing Marginal Cost efficiently in terms of local computation and message complexity. The bit complexity (per edge) of this algorithm depends linearly on the product of the tree height and the number of possible rates. However, we provide two lower bounds on bit complexity, demonstrating that computing Marginal Cost (a) with multiple rates requires a bit complexity that is linear in the number of rates, and (b) with a cost for enabling multicasting requires a bit complexity that is linear in the height of the tree. A modification of our algorithm for the layered paradigm also applies to the split session paradigm of providing multiple rates, but in this case, both the local computation and the bit complexity become exponential in the number of possible rates. However, we also demonstrate that for the split session paradigm, the problem becomes NP-Hard even to approximate if the number of possible rates is part of the input. This indicates that we cannot expect to do much better than the algorithm we provide. Finally, we examine the effect of delivering the information for the different rates from different locations within the network. We show that in this case, the Marginal Cost problem becomes NP-Hard in the split session paradigm even for a constant number of possible rates, but that in the layered paradigm it can be solved in polynomial time by formulating the problem as a linear program that is guaranteed to have an integral optimal solution.