The approximation threshold for metric $k$-median: is $1+2/e$ attainable?

Theorem 1.1: for every fixed $\varepsilon>0$, a deterministic algorithm running in time polynomial in the binary input length (the polynomial depends on $\varepsilon$) returns $S\subseteq F$ with $|S|\le k$ and cost at most $(1+2/e+\varepsilon)\mathrm{OPT}_k$, for arbitrary $k$ and arbitrary finite rational metrics with specified candidate facilities. Corollary 1.2: under $P\ne NP$, the infimum of polynomial-time approximation factors in this model is $1+2/e$, the lower bound being the known coverage gap. It does not give a $(1+2/e)$-approximation itself, does not settle the model where every client is a candidate facility ($F=J$), whose lower bound differs, and claims no practical running time. The companion proves a randomized $(2-\sigma)$-approximation for an absolute $\sigma>0$ by anchor recovery and bounded-price payment accounting.

Theorem 1.1: for every fixed $\varepsilon>0$, a deterministic algorithm running in time polynomial in the binary input length (the polynomial depends on $\varepsilon$) returns $S\subseteq F$ with $|S|\le k$ and cost at most $(1+2/e+\varepsilon)\mathrm{OPT}_k$, for arbitrary $k$ and arbitrary finite rational metrics with specified candidate facilities. Corollary 1.2: under $P\ne NP$, the infimum of polynomial-time approximation factors in this model is $1+2/e$, the lower bound being the known coverage gap. It does not give a $(1+2/e)$-approximation itself, does not settle the model where every client is a candidate facility ($F=J$), whose lower bound differs, and claims no practical running time. The companion proves a randomized $(2-\sigma)$-approximation for an absolute $\sigma>0$ by anchor recovery and bounded-price payment accounting.