1. A method for operating a first path computation element in a first autonomous system to participate in establishing a Multi-Protocol Label Switching (MPLS) Traffic Engineering Label Switched Path (LSP) from a head-end node to a tail-end node, said first autonomous system being positioned between a second autonomous system and a third autonomous system, each of said first, second, and third autonomous systems comprising a group of routers subject to a common authority, said method comprising: receiving at said first path computation element, a path computation request forwarded from a second path computation element in said second autonomous system; forwarding from said first path computation element, said path computation request to a third path computation element in said third autonomous system; and upon receiving said path computation request and prior to receiving a response from said third path computation element, computing constraints-satisfying shortest path segments between border routers coupling said first autonomous system to said second autonomous system and border routers coupling said first autonomous system and said third autonomous system using constraints received in said path computation request.