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Let $(\{1,2,\ldots,n\},d)$ be a metric space. We analyze the expected valueand the variance of $\sum_{i=1}^{\lfloorn/2\rfloor}\,d({\boldsymbol{\pi}}(2i-1),{\boldsymbol{\pi}}(2i))$ for auniformly random permutation ${\boldsymbol{\pi}}$ of $\{1,2,\ldots,n\}$,leading to the following results: (I) Consider the problem of finding a pointin $\{1,2,\ldots,n\}$ with the minimum sum of distances to all points. We showthat this problem has a randomized algorithm that (1) always outputs a$(2+\epsilon)$-approximate solution in expected $O(n/\epsilon^2)$ time and that(2) inherits Indyk's~\cite{Ind99, Ind00} algorithm to output a$(1+\epsilon)$-approximate solution in $O(n/\epsilon^2)$ time with probability$\Omega(1)$, where $\epsilon\in(0,1)$. (II) The average distance in$(\{1,2,\ldots,n\},d)$ can be approximated in $O(n/\epsilon)$ time to within amultiplicative factor in $[\,1/2-\epsilon,1\,]$ with probability$1/2+\Omega(1)$, where $\epsilon>0$. (III) Assume $d$ to be a graph metric.Then the average distance in $(\{1,2,\ldots,n\},d)$ can be approximated in$O(n)$ time to within a multiplicative factor in $[\,1-\epsilon,1+\epsilon\,]$with probability $1/2+\Omega(1)$, where $\epsilon=\omega(1/n^{1/4})$.

Metric random matchings with applications