Fitting Tree Metrics with Minimum Disagreements

TL;DR

This paper presents an optimal constant-approximation algorithm for the $L_0$ Fitting Tree Metrics problem, extending previous ultrametric solutions to tree metrics with a blackbox approach, addressing a known computational hardness.

Contribution

It introduces a framework that extends existing ultrametric approximation algorithms to tree metrics, achieving an asymptotically optimal solution for $L_0$ Fitting Tree Metrics.

Findings

Achieves an $O(1)$ approximation for $L_0$ Fitting Tree Metrics.

Extends ultrametric approximation techniques to general tree metrics.

Provides a blackbox method to convert ultrametric solutions into tree metric solutions.

Abstract

In the $L_0$ Fitting Tree Metrics problem, we are given all pairwise distances among the elements of a set $V$ and our output is a tree metric on $V$. The goal is to minimize the number of pairwise distance disagreements between the input and the output. We provide an $O(1)$ approximation for $L_0$ Fitting Tree Metrics, which is asymptotically optimal as the problem is APX-Hard. For $p\ge 1$, solutions to the related $L_p$ Fitting Tree Metrics have typically used a reduction to $L_p$ Fitting Constrained Ultrametrics. Even though in FOCS '22 Cohen-Addad et al. solved $L_0$ Fitting (unconstrained) Ultrametrics within a constant approximation factor, their results did not extend to tree metrics. We identify two possible reasons, and provide simple techniques to circumvent them. Our framework does not modify the algorithm from Cohen-Addad et al. It rather extends any $\rho$ approximation for $L_0$ Fitting Ultrametrics to a $6\rho$ approximation for $L_0$ Fitting Tree Metrics in a blackbox fashion.