A technique efficiently routes Internet Protocol (IP) traffic on paths between customer edge devices (CEs) across a provider network ("CE-CE paths") in a computer network. According to the novel technique, a path computation element (PCE), e.g., a provider edge device (PE), may learn dynamic link attribute information of remote links from the provider network to one or more remote CEs (e.g., "PE-CE links" or "CE-PE links"). A multi-homed requesting CE requests from the PCE a set of CE-CE path metrics (e.g., costs) to one or more remote destination address prefixes, e.g., via each multi-homed CE-PE link from the requesting CE. In response to the request, the PCE computes the set of available CE-CE paths and current metrics to the remote destination address prefixes and returns the corresponding CE-CE path metrics to the requesting CE. The requesting CE modifies its IP forwarding entries accordingly in order to perform IP traffic routing corresponding to the CE-CE path metrics (e.g., asymmetrical load balancing) across its multi-homed CE-PE links.