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A packet network of interconnected nodes employs a method of routing with service-level guarantees to determine a path through the network for a requested multicast, label-switched path Each of the nodes includes one or more routers that forward packets based on a forwarding table constructed from a directed tree determined in accordance with the method of multicast routing with service-level guarantees. For a first implementation, a heuristic algorithm uses a scaling phase that iteratively adjusts a maximum arc capacity, determines the resulting tree for the iteration, and selects the tree as the routing tree that provides the “maximum” flow. For a second implementation, the heuristic algorithm computes maximum multicast flows and determines links in the network that are “critical” to satisfy future multicast routing requests. A multicast routing tree is selected such that provisioning the flows over its links “minimally …