1. A method for a computing system of a program execution service to execute programs in locations selected in accordance with user-specified constraints, the program execution service using a plurality of host computing systems in multiple geographic areas to execute programs, the plurality of host computing systems being connected at multiple hierarchical levels that are each associated with a distinct degree of proximity such that computing systems of a group at a first hierarchical level are physically proximate and share a common group data exchange medium that is specific to that group and such that each successive higher hierarchical level includes computing system that are less proximate and connected via one or more other data exchange mediums, the method comprising: receiving an indication from a user of the program execution service to execute an indicated quantity of copies of an indicated program in such a manner as to satisfy one or more indicated execution constraints, the one or more execution constraints for use by the program execution service in selecting one or more of the plurality of host computing systems to execute the indicated quantity of indicated program copies, the one or more execution constraints specifying one or more degrees of proximity of the selected one or more host computing systems to one or more indicated other target computing systems; automatically determining one or more of the hierarchical levels whose associated degrees of proximity correspond to the one or more degrees of proximity specified by the one or more execution constraints; automatically determining one or more candidate host computing systems that are available to execute the indicated quantity of indicated program copies in accordance with the one or more execution constraints, the candidate host computing systems being connected to the indicated target computing systems at one or more of the determined hierarchical levels; automatically selecting from the candidate host computing systems the one or more host computing systems to execute the indicated quantity of indicated program copies, such that each of the selected one or more host computing systems is to execute at least one of the indicated program copies; and initiating executing of the indicated quantity of indicated program copies on the selected one or more host computing systems.