Approximating the Median under the Ulam Metric

TL;DR

This paper develops new approximation algorithms for the median string problem under the Ulam metric, surpassing the known 2-approximation barrier and addressing both worst-case and probabilistic models.

Contribution

It introduces algorithms that achieve better than 2-approximation for the median under the Ulam metric, including in specific cases and probabilistic models.

Findings

Breaks the 2-approximation barrier with a (2-δ)-approximate algorithm.

Provides a (2-δ)-approximation for median strings with large objective value.

Designs a high-probability (1+o(1/ε))-approximate median algorithm for perturbed permutations.

Abstract

We study approximation algorithms for variants of the \emph{median string} problem, which asks for a string that minimizes the sum of edit distances from a given set of $m$ strings of length $n$. Only the straightforward $2$-approximation is known for this NP-hard problem. This problem is motivated e.g.~by computational biology, and belongs to the class of median problems (over different metric spaces), which are fundamental tasks in data analysis. Our main result is for the Ulam metric, where all strings are permutations over $[n]$ and each edit operation moves a symbol (deletion plus insertion). We devise for this problem an algorithms that breaks the $2$-approximation barrier, i.e., computes a $(2-\delta)$-approximate median permutation for some constant $\delta>0$ in time $\tilde{O}(nm^2+n^3)$. We further use these techniques to achieve a $(2-\delta)$ approximation for the median string problem in the special case where the median is restricted to length $n$ and the optimal objective is large $\Omega(mn)$. We also design an approximation algorithm for the following probabilistic model of the Ulam median: the input consists of $m$ perturbations of an (unknown) permutation $x$, each generated by moving every symbol to a random position with probability (a parameter) $\epsilon>0$. Our algorithm computes with high probability a $(1+o(1/\epsilon))$-approximate median permutation in time $O(mn^2+n^3)$.