Clustering Permutations: New Techniques with Streaming Applications

Abstract: We study the classical metric $k$-median clustering problem over a set of input rankings (i.e., permutations), which has myriad applications, from social-choice theory to web search and databases. A folklore algorithm provides a $2$-approximate solution in polynomial time for all $k=O(1)$, and works irrespective of the underlying distance measure, so long it is a metric; however, going below the $2$-factor is a notorious challenge. We consider the Ulam distance, a variant of the well-known edit-distance metric, where strings are restricted to be permutations. For this metric, Chakraborty, Das, and Krauthgamer [SODA, 2021] provided a $(2-\delta)$-approximation algorithm for $k=1$, where $\delta\approx 2<sup>{-40}$. Our primary contribution is a new algorithmic framework for clustering a set of permutations. Our first result is a $1.999$-approximation algorithm for the metric $k$-median problem under the Ulam metric, that runs in time $(k \log (nd))^{O(k)}n d<sup>3$ for an input consisting of $n$ permutations over $[d]$. In fact, our framework is powerful enough to extend this result to the streaming model (where the $n$ input permutations arrive one by one) using only polylogarithmic (in $n$) space. Additionally, we show that similar results can be obtained even in the presence of outliers, which is presumably a more difficult problem.