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Ordinally Consensus Subset over Multiple Metrics
arXiv - CS - Computational Complexity Pub Date : 2020-11-11 , DOI: arxiv-2011.05557
Dingkang Wang, Yusu Wang

In this paper, we propose to study the following maximum ordinal consensus problem: Suppose we are given a metric system (M, X), which contains k metrics M = {\rho_1,..., \rho_k} defined on the same point set X. We aim to find a maximum subset X' of X such that all metrics in M are "consistent" when restricted on the subset X'. In particular, our definition of consistency will rely only on the ordering between pairwise distances, and thus we call a "consistent" subset an ordinal consensus of X w.r.t. M. We will introduce two concepts of "consistency" in the ordinal sense: a strong one and a weak one. Specifically, a subset X' is strongly consistent means that the ordering of their pairwise distances is the same under each of the input metric \rho_i from M. The weak consistency, on the other hand, relaxes this exact ordering condition, and intuitively allows us to take the plurality of ordering relation between two pairwise distances. We show in this paper that the maximum consensus problems over both the strong and the weak consistency notions are NP-complete, even when there are only 2 or 3 simple metrics, such as line metrics and ultrametrics. We also develop constant-factor approximation algorithms for the dual version, the minimum inconsistent subset problem of a metric system (M, P), - note that optimizing these two dual problems are equivalent.

中文翻译:

多个指标的有序共识子集

在本文中,我们建议研究以下最大序数共识问题:假设我们给定一个度量系统 (M, X),其中包含定义在同一点上的 k 个度量 M = {\rho_1,..., \rho_k}集合 X。我们的目标是找到 X 的最大子集 X',使得当限制在子集 X' 上时,M 中的所有度量都是“一致的”。特别是,我们对一致性的定义将仅依赖于成对距离之间的排序,因此我们将“一致”子集称为 X wrt M 的序数共识。 我们将介绍序数意义上的“一致性”的两个概念:强一个和一个弱的。具体来说,子集 X' 是强一致性意味着它们的成对距离的排序在来自 M 的每个输入度量 \rho_i 下是相同的。另一方面,弱一致性,放宽了这种精确排序条件,直观地允许我们采用两个成对距离之间的多个排序关系。我们在本文中表明,强一致性和弱一致性概念上的最大共识问题都是 NP 完全的,即使只有 2 或 3 个简单度量,例如线度量和超度量。我们还为对偶版本开发了常数因子逼近算法,即度量系统 (M, P) 的最小不一致子集问题,请注意优化这两个对偶问题是等效的。即使只有 2 或 3 个简单指标,例如线指标和超指标。我们还为对偶版本开发了常数因子逼近算法,即度量系统 (M, P) 的最小不一致子集问题,请注意优化这两个对偶问题是等效的。即使只有 2 或 3 个简单指标,例如线指标和超指标。我们还为对偶版本开发了常数因子逼近算法,即度量系统 (M, P) 的最小不一致子集问题,请注意优化这两个对偶问题是等效的。
更新日期:2020-11-12
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