Metric random matchings with applications

Let $(\{1,2,\ldots,n\},d)$ be a metric space. We analyze the expected value and the variance of $\sum_{i=1}^{\lfloor n/2\rfloor}\,d({\boldsymbolπ}(2i-1),{\boldsymbolπ}(2i))$ for a uniformly random permutation ${\boldsymbolπ}$ of $\{1,2,\ldots,n\}$, leading to the following results: (I) Consider the problem of finding a point in $\{1,2,\ldots,n\}$ with the minimum sum of distances to all points. We show that this problem has a randomized algorithm that (1) always outputs a $(2+ε)$-approximate solution in expected $O(n/ε^2)$ time and that (2) inherits Indyk's~\cite{Ind99, Ind00} algorithm to output a $(1+ε)$-approximate solution in $O(n/ε^2)$ time with probability $Ω(1)$, where $ε\in(0,1)$. (II) The average distance in $(\{1,2,\ldots,n\},d)$ can be approximated in $O(n/ε)$ time to within a multiplicative factor in $[\,1/2-ε,1\,]$ with probability $1/2+Ω(1)$, where $ε>0$. (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 $[\,1-ε,1+ε\,]$ with probability $1/2+Ω(1)$, where $ε=ω(1/n^{1/4})$.