Cross-Monotonic Multicast

In the routing and cost sharing of multicast towards a group of potential receivers, cross-monotonicity is a property that states a user's payment can only be smaller when serviced in a larger set. Being cross-monotonic has been shown to be the key in achieving group-strategyproofness. We study multicast schemes that target optimal flow routing, cross-monotonic cost sharing, and budget balance. We show that no multicast scheme can satisfy these three properties simultaneously, and resort to approximate budget balance instead. We derive both positive and negative results that complement each other for directed and undirected networks. We show that in directed networks, no cross-monotonic scheme can recover a constant fraction of optimal multicast cost. We provide a simple scheme that does achieve 1/k-budget-balance, where k is the number of receivers. Using a probabilistic method rooted in random graph theory, we prove an upper-bound of 2/radic(k) for the budget balance ratio. For undirected networks, we derive a constant upper-bound of 1/2 instead. We further apply a smooth dual growing technique to design a cross- monotonic scheme that recovers k+1/2kzeta of optimal multicast cost in undirected networks, where zeta is a network-dependent parameter close to 1. This is almost tight against the upper-bound |. We finally present a two-stage linear optimization model that pursues maximum budget balance in any given specific network, with trade-off in complexity. Optimization results in various network configurations confirm the theoretically established bounds.