Fitting Metrics and Ultrametrics with Minimum Disagreements

Given $x\in(\mathbb{R}_{\geqslant 0})(_{2}^{[n]})$ recording pairwise distances, the Metric Violation Distance problem asks to compute the $\ell_{0}$ distance between x and the metric cone; i.e., modify the minimum number of entries of x to make it a metric. Due to its large number of applications in various data analysis and optimization tasks, this problem has been actively studied recently. We present an $O(\log n)$-approximation algorithm for METRIC VIOLATION Distance, exponentially improving the previous best approximation ratio of $O(OPT^{1/3})$ of Fan, Raichel, and Van Buskirk [SODA, 2018]. Furthermore, a major strength of our algorithm is its simplicity and running time. We also study the related problem of Ultrametric Violation Distance, where the goal is to compute the $\ell_{0}$ distance to the cone of ultrametrics, and achieve a constant factor approximation algorithm. The ULTRAMETRIC VIOLATION DISTANCE problem can be regarded as an extension of the problem of fitting ultrametrics studied by Ailon and Charikar [SIAM J. Computing, 2011] and by Cohen-Addad, Das, Kipouridis, Parotsidis, and Thorup [FOCS, 2021] from $\ell_{1}$ norm to $\ell_{0}$ norm. We show that this problem can be favorably interpreted as an instance of CORRELATION CLUSTERING with an additional hierarchical structure, which we solve using a new $O(1)$-approximation algorithm for correlation clustering that has the structural property that it outputs a refinement of the optimum clusters. An algorithm satisfying such a property can be considered of independent interest. We also provide an $O(\log n\log\log n)$ approximation algorithm for weighted instances. Finally, we investigate the complementary version of these problems where one aims at choosing a maximum number of entries of x forming an (ultra-)metric. In stark contrast with the minimization versions, we prove that these maximization versions are hard to approximate within any constant factor assuming the Unique Games Conjecture.