We study the natural problem of Triplet Reconstruction (also Rooted Triplets Consistency or Triplet Clustering), originally motivated in computational biology and relational databases (Aho, Sagiv, Szymanski, and Ullman, 1981): given $n$ points, we want to embed them onto the $n$ leaves of a rooted binary tree (a hierarchical clustering or ultrametric embedding) such that a given set of $m$ triplet constraints is satisfied. Triplet $ij|k$ indicates that ``$i, j$ are more closely related to each other than to $k$'' and a tree satisfies $ij|k$ if $d(i,j)$ is the smallest among the 3 distances. Aho et al. (1981) gave an elegant efficient algorithm to find a tree respecting all constraints (if it exists) and it is easy to see that a random binary tree is a 1/3-approximation. Unfortunately, despite more than four decades of research, no better approximation is known. Our main theorem--which captures Triplet Reconstruction as a special case--is a general hardness of approximation result about Constraint Satisfaction Problems (CSPs) over infinite domains (the variables are mapped to any of the $n$ leaves of a tree). Specifically, we prove, under Unique Games (Khot, 2002), that Triplet Reconstruction and more generally, every CSP over hierarchies is approximation resistant (there is no polynomial-time algorithm that does asymptotically better than a biased random assignment). This settles the approximability for many interesting Subtree or Supertree Aggregation Problems. More broadly, our result significantly extends the list of approximation resistant predicates and is a generalization of Guruswami, Hastad, Manokaran, Raghavendra, and Charikar (2011), who showed that ordering CSPs are approximation resistant. The main challenge in our analyses stems from the fact that trees have topology which is what determines whether a given triplet constraint on the leaves is satisfied or not.

翻译：我们研究三重态重建的自然问题（亦称有根三重态一致性或三重态聚类），该问题最初源于计算生物学和关系数据库（Aho, Sagiv, Szymanski, Ullman, 1981）：给定 $n$ 个点，需将其嵌入到有根二叉树的 $n$ 个叶子节点（层次聚类或超度量嵌入），使得给定的 $m$ 个三重态约束得到满足。三重态 $ij|k$ 表示"$i,j$ 之间的关系比它们与 $k$ 的关系更近"，当 $d(i,j)$ 是三个距离中最小时，称树满足 $ij|k$。Aho 等人（1981）提出了一个优雅的高效算法以寻找满足所有约束的树（若存在），且容易看出随机二叉树的近似比为 1/3。遗憾的是，历经四十余年的研究，仍未有更好的近似比。我们的主要定理——将三重态重建作为特例——是关于无限域上约束满足问题的通用近似困难结果（变量被映射到树的任意 $n$ 个叶子节点）。具体而言，我们在唯一博弈假设（Khot, 2002）下证明，三重态重建以及更一般地，所有层级结构上的约束满足问题都是近似困难的（不存在渐近优于有偏随机赋值的多项式时间算法）。这解决了众多有趣的子树或超树聚合问题的可近似性。更广泛地，我们的结果显著扩展了近似困难谓词的列表，并推广了 Guruswami, Hastad, Manokaran, Raghavendra, Charikar（2011）关于排序约束满足问题近似困难的结论。我们分析中的主要挑战源于树的拓扑结构——它决定了叶子节点上的给定三重态约束是否得到满足。