In this paper, we study a number of well-known combinatorial optimization problems that fit in the following paradigm: the input is a collection of (potentially inconsistent) local relationships between the elements of a ground set (e.g., pairwise comparisons, similar/dissimilar pairs, or ancestry structure of triples of points), and the goal is to aggregate this information into a global structure (e.g., a ranking, a clustering, or a hierarchical clustering) in a way that maximizes agreement with the input. Well-studied problems such as rank aggregation, correlation clustering, and hierarchical clustering with triplet constraints fall in this class of problems. We study these problems on stochastic instances with a hidden embedded ground truth solution. Our main algorithmic contribution is a unified technique that uses the maximum cut problem in graphs to approximately solve these problems. Using this technique, we can often get approximation guarantees in the stochastic setting that are better than the known worst case inapproximability bounds for the corresponding problem. On the negative side, we improve the worst case inapproximability bound on several hierarchical clustering formulations through a reduction to related ranking problems.

中文翻译：

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通过MAX-CUT最大化排名，聚类和分层聚类的协议

在本文中，我们研究了一些符合以下范式的众所周知的组合优化问题：输入是基础集元素之间（可能成对不一致）局部关系的集合（例如，成对比较，相似/不相似）对，或三元组的祖先结构），目标是以最大程度地与输入达成一致的方式将此信息聚合为全局结构（例如，排名，聚类或分层聚类）。在这类问题中，经过深入研究的问题，例如等级聚合，相关性聚类和具有三重态约束的层次聚类。我们使用隐藏的嵌入式地面事实解决方案在随机实例上研究这些问题。我们的主要算法贡献是一种统一的技术，该技术使用图形中的最大割问题来近似解决这些问题。使用这种技术，我们经常可以在随机环境中获得比相应问题的已知最坏情况下不可逼近范围更好的逼近保证。消极的一面，我们通过减少相关排名问题，改善了几种层次聚类公式的最坏情况的不可逼近性。