New AI Math Could Change Where Your Next Store Opens

A new paper designs randomized strategyproof mechanisms for multi-facility location, aiming to minimize the sum of agents’ distances to their nearest facilities. For two facilities, a mechanism called Pairwise-Distance is strategyproof on Ptolemaic spaces—including Euclidean and Hilbert spaces—and achieves an approximation ratio of 4. A hybrid of this with the classical Proportional mechanism is also strategyproof on Ptolemaic spaces and has a tight approximation ratio of about 3.5186, breaking the long-standing factor-4 benchmark. For settings with many facilities, the paper introduces new mechanisms: one works on any metric space and improves the previous best-known ratio for a certain case, while another works on the line for three facilities with a constant 6-approximation—replacing an earlier n-dependent guarantee—but is not strategyproof for four or more facilities.