We give an exact-budget recovery algorithm for metric k-median with single-exponential dependence on the number of comparison clusters without accurate, distinct proxies in a supplied anchor solution. On positive integral metrics of polynomially bounded diameter, a sufficiently small total proxy error and logarithmically many such clusters yield a $(1+2/e+\varepsilon)$ approximation in polynomial time with arbitrarily high success probability. We also prove bounded-price strictness for one compatible execution of the logarithmic-surplus construction. Together the recovery and payment arguments give a randomized $(2-\sigma)$ approximation, for an absolute σ > 0, on arbitrary finite rational metrics, both with high probability and in expectation, while opening at most k facilities on every output.

For every fixed ε > 0, we give a deterministic polynomial-time $(1+2/e+\varepsilon)$-approximation for finite rational metric k-median with specified candidate facilities, opening at most k facilities. Under $P\ne NP$, the infimum approximation factor in this model is therefore $1+2/e$.