Business relationships between ASes in the Internet are typically confidential, yet knowledge of them is essential to understand many aspects of Internet structure, performance, dynamics, and evolution. We present a new algorithm to infer these relationships using BGP paths. Unlike previous approaches, our algorithm does not assume the presence (or seek to maximize the number) of valley-free paths, instead relying on three assumptions about the Internet's inter-domain structure: (1) an AS enters into a provider relationship to become globally reachable; and (2) there exists a peering clique of ASes at the top of the hierarchy, and (3) there is no cycle of p2c links. We assemble the largest source of validation data for AS-relationship inferences to date, validating 34.6% of our 126,082 c2p and p2p inferences to be 99.6% and 98.7% accurate, respectively. Using these inferred relationships, we evaluate three algorithms for inferring each AS's customer cone, defined as the set of ASes an AS can reach using customer links. We demonstrate the utility of our algorithms for studying the rise and fall of large transit providers over the last fifteen years, including recent claims about the flattening of the AS-level topology and the decreasing influence of "tier-1" ASes on the global Internet.