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.

翻译：在最小分歧拟合树度量问题中，给定集合 $V$ 中元素之间的所有成对距离，输出是 $V$ 上的一个树度量。目标是使输入与输出之间成对距离分歧的数量最小化。我们为最小分歧拟合树度量问题提供了 $O(1)$ 近似比，这是渐近最优的，因为该问题是APX-Hard的。对于 $p\ge 1$，相关的最小分歧拟合树度量问题的解通常通过归约为最小分歧拟合约束超度量问题得到。尽管在FOCS '22会议上，Cohen-Addad等人以常数近似因子解决了最小分歧拟合（无约束）超度量问题，但他们的结果并未扩展到树度量。我们识别了两个可能的原因，并提供了简单的技巧来规避它们。我们的框架并未修改Cohen-Addad等人的算法，而是以黑盒方式将任何 $\rho$ 近似的最小分歧拟合超度量算法扩展为 $6\rho$ 近似的最小分歧拟合树度量算法。