Robust-Sorting and Applications to Ulam-Median

2025-02-11

AI Summary

This paper studies robust comparison sorting under adversarial noise: given that at most |B| pairwise comparisons may be arbitrarily corrupted by a malicious adversary, how can one achieve approximate sorting with low query complexity? We propose an efficient algorithm based on randomized pivot selection and tournament graph modeling. To our knowledge, this is the first algorithm achieving expected $ ilde{O}(n)$ query complexity in the bounded-adversary model, while guaranteeing Ulam distance distortion at most $(3+varepsilon)|B|$. Furthermore, we improve the time complexity of the fixed-parameter tractable (FPT) approximation algorithm for the Ulam-$k$-Median problem from superlinear to linear, while preserving a $(2-varepsilon)$-approximation ratio—yielding the first linear-time $(2-varepsilon)$-approximation algorithm for this problem.

Technology Category

• Search and Optimization: Adversarial Search
• Machine Learning: Adversarial Learning & Robustness
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Sorting is one of the most basic primitives in many algorithms and data analysis tasks. Comparison-based sorting algorithms, like quick-sort and merge-sort, are known to be optimal when the outcome of each comparison is error-free. However, many real-world sorting applications operate in scenarios where the outcome of each comparison can be noisy. In this work, we explore settings where a bounded number of comparisons are potentially corrupted by erroneous agents, resulting in arbitrary, adversarial outcomes. We model the sorting problem as a query-limited tournament graph where edges involving erroneous nodes may yield arbitrary results. Our primary contribution is a randomized algorithm inspired by quick-sort that, in expectation, produces an ordering close to the true total order while only querying $ ilde{O}(n)$ edges. We achieve a distance from the target order $pi$ within $(3 + epsilon)|B|$, where $B$ is the set of erroneous nodes, balancing the competing objectives of minimizing both query complexity and misalignment with $pi$. Our algorithm needs to carefully balance two aspects: identify a pivot that partitions the vertex set evenly and ensure that this partition is"truthful"and yet query as few"triangles"in the graph $G$ as possible. Since the nodes in $B$ can potentially hide in an intricate manner, our algorithm requires several technical steps. Additionally, we demonstrate significant implications for the Ulam-$k$-Median problem, a classical clustering problem where the metric is defined on the set of permutations on a set of $d$ elements. Chakraborty, Das, and Krauthgamer gave a $(2-varepsilon)$ FPT approximation algorithm for this problem, where the running time is super-linear in both $n$ and $d$. We use our robust sorting framework to give the first $(2-varepsilon)$ FPT linear time approximation algorithm for this problem.

Problem

• Addresses sorting with noisy comparisons.
• Develops robust algorithm for adversarial scenarios.
• Improves Ulam-k-Median approximation efficiency.

Innovation

• Randomized quick-sort inspired algorithm
• Query-limited tournament graph model
• Linear time Ulam-k-Median approximation