We design and analyze randomized strategyproof mechanisms for multi-facility location under the utilitarian social-cost objective, the sum of the agents' distances to their nearest facilities. For two facilities, the Pairwise-Distance mechanism locates facilities at a pair of reported locations sampled with probability proportional to their distance. It is strategyproof on Ptolemaic spaces, including Euclidean and Hilbert spaces as special cases, and has an approximation ratio of \(4\). The resulting Hybrid-Distance mechanism is a fixed-probability mixture: it selects the classical Proportional mechanism [Lu et al., EC'10] with probability \(λ^*=\frac{5+4\sqrt3}{23}\) and Pairwise-Distance with probability $1-λ^*$. It is strategyproof on Ptolemaic spaces and has a tight approximation ratio of \(\frac{74+4\sqrt3}{23}\approx3.5186\), breaking the long-standing factor-\(4\) benchmark of [Lu et al., EC'10]. We complement the two-facility results by studying more facilities. First, for \(n\) agents and \(k=n-1\) facilities, we introduce the Inverse-Square mechanism, which omits one report with probability proportional to the inverse square of its nearest-neighbor distance and locates facilities at all remaining reports. It is strategyproof on any metric space and has an approximation ratio of \(Θ(\sqrt{n})\), improving the previous best-known ratio of \(\frac{n}{2}\) [Escoffier et al., ADT'11]. Second, for $k$ facilities on the line, we introduce the Gap-Product mechanism, which locates facilities at \(k\) reports and weights each set by the product of the gaps between consecutive selected reports. When \(k=3\), it is strategyproof and has a \(6\)-approximation, replacing the previous \(n\)-dependent guarantee [Fotakis and Tzamos, EC'13] by a constant, whereas it is not strategyproof for any \(k\ge4\).

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