Sorting from Counterexamples

2026-08-21

AI Summary

研究了在最多k个错误反馈情况下，通过最少查询次数学习n项未知线性排序的问题，提出基于几何表示和Grünbaum定理的上界解法。

Abstract

Consider the following problem of learning an unknown linear order on $n$ items. In each round, the learner guesses a complete ordering of the items and receives either confirmation that the guess is correct or a counterexample: a pair of items in the wrong order. The goal is to identify the unknown order using as few queries as possible. We study this problem when up to $k$ of the returned counterexamples may be untruthful, where $k$ is not known in advance. We determine the optimal query complexity up to constant factors: \[ Θ(n\log n + nk). \] Thus, while the noiseless complexity matches the classical complexity of sorting, each untruthful counterexample incurs an additional cost of order $n$. The upper bound is based on a geometric representation of permutations and Grünbaum's theorem, while the lower bound combines sorting arguments with a Condorcet-type construction. We also study the case where the target ranking has a low-dimensional geometric representation: each item is represented by a point in $\mathbb{R}^d$, and the ranking is obtained by projecting the points onto an unknown direction. For these classes we give an upper bound of $O(d^2\log n+dk)$ and a lower bound of $Ω(d\log n+dk)$, leaving a factor of $d$ gap in the noiseless term.

Problem

• linear order
• counterexamples
• untruthful feedback
• query complexity

Innovation

• counterexamples
• query complexity
• geometric representation
• Grünbaum's theorem
• low-dimensional geometric representation