Brief Announcement: On the Hardness of Topology Inference

Many systems require information about the topology of networks on the Internet, for purposes like management, efficiency, testing of new protocols and so on. However, ISPs usually do not share the actual topology maps with outsiders. Consequently, many systems have been devised to reconstruct the topology of networks on the Internet from publicly observable data. Such systems rely on traceroute to provide path information, and attempt to compute the network topology from these paths. However, traceroute has the problem that some routers refuse to reveal their addresses, and appear as anonymous nodes in traces. Previous research on the problem of topology inference with anonymous nodes has demonstrated that it is at best NP-complete. We prove a stronger result. There exists no algorithm that, given an arbitrary trace set with anonymous nodes, can determine the topology of the network that generated the trace set. Even the weak version of the problem, which allows an algorithm to output a “small” set of topologies such that the correct topology is included in the solution set, is not solvable: there exist trace sets such that any algorithm guaranteed to output the correct topology outputs at least an exponential number of networks. We show how to construct such a pathological case even when the network is known to have exactly two anonymous nodes.