We study the Ultrametric Violation Distance problem introduced by Cohen-Addad, Fan, Lee, and Mesmay [FOCS, 2022]. Given pairwise distances $x\in \mathbb{R}_{>0}^{\binom{[n]}{2}}$ as input, the goal is to modify the minimum number of distances so as to make it a valid ultrametric. In other words, this is the problem of fitting an ultrametric to given data, where the quality of the fit is measured by the $\ell_0$ norm of the error; variants of the problem for the $\ell_\infty$ and $\ell_1$ norms are well-studied in the literature. Our main result is a 5-approximation algorithm for Ultrametric Violation Distance, improving the previous best large constant factor ($\geq 1000$) approximation algorithm. We give an $O(\min\{L,\log n\})$-approximation for weighted Ultrametric Violation Distance where the weights satisfy triangle inequality and $L$ is the number of distinct values in the input. We also give a $16$-approximation for the problem on $k$-partite graphs, where the input is specified on pairs of vertices that form a complete $k$-partite graph. All our results use a unified algorithmic framework with small modifications for the three cases.

翻译：我们研究了由Cohen-Addad、Fan、Lee和Mesmay [FOCS, 2022]提出的超度量违反距离问题。给定成对距离 $x\in \mathbb{R}_{>0}^{\binom{[n]}{2}}$ 作为输入，目标是修改最少数量的距离，使其成为一个有效的超度量。换言之，这是将超度量拟合到给定数据的问题，其中拟合质量由误差的 $\ell_0$ 范数衡量；该问题在 $\ell_\infty$ 和 $\ell_1$ 范数下的变体已在文献中得到充分研究。我们的主要结果是针对超度量违反距离的一个5-近似算法，改进了此前最佳的大常数因子（$\geq 1000$）近似算法。对于带权重的超度量违反距离（其中权重满足三角不等式，且 $L$ 是输入中不同值的个数），我们给出了一个 $O(\min\{L,\log n\})$ 的近似算法。此外，针对 $k$-部图上的问题（输入由构成完全 $k$-部图的顶点对指定），我们给出了一个16-近似算法。我们所有的结果均采用统一的算法框架，仅针对三种情况进行了微小调整。