The metric k-median approximation threshold and recovery

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+ε) 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.