How to Fix a Broken Metric: A Linear Kernel, Tight Bounds, and Tree Metrics

2026-10-05

AI Summary

This study addresses the metric violation distance problem in noisy data repair, aiming to overcome the efficiency bottlenecks of parameterized algorithms and establish tight bounds. By employing parameterized complexity theory, kernelization techniques, ETH-based lower bound analysis, and interval generalization strategies, this work proposes the first single-exponential-time algorithm running in $2^{O(k)}$ for general metrics, along with a linear vertex kernel of size $6k$. It further establishes the fixed-parameter tractability of the problem under tree metrics. Additionally, the study determines optimal time complexity lower bounds and achieves an improved logarithmic approximation ratio without additional computational overhead. Collectively, these contributions systematically resolve several open problems in this research area.

Abstract

Given a complete graph whose edge weights represent dissimilarities, Metric Violation Distance asks whether at most $k$ weights can be changed to form a metric. Motivated by metric repair for noisy data, the problem admits a polynomial-time $O(\log n)$-approximation due to Cohen-Addad, Fan, Lee and de Mesmay [SIAM J. Comput., 2025]. In the context of parameterized complexity, Fan, Gilbert, Raichel, Sonthalia and Van Buskirk [SWAT 2020] gave a $k^{O(k)}n^{O(1)}$-time algorithm. Fomin, Golovach and More [IPEC 2026] obtained an $O(k^2)$ kernel and a single-exponential algorithm for the ultrametric case. They asked whether general metrics admit a single-exponential algorithm and a polynomial kernel, and whether the tree-metric analogue is fixed-parameter tractable. We answer all three questions. We give a $2^{O(k)}n^{O(1)}$-time algorithm and prove that, unless ETH fails, no $2^{o(k)}n^{O(1)}$-time algorithm exists, even when all input distances lie in $\{1,2,3\}$. We also give a kernel with at most $6k$ vertices. The kernel supports solution lifting and can precede any approximation algorithm. Combined with the $O(\log n)$-approximation of Cohen-Addad, Fan, Lee and de Mesmay, it yields an $O(\log \mathrm{OPT})$-approximation at no asymptotic cost in running time. Both results extend to an interval generalization in which each edge $e$ has an observed value $M_e$ and an admissible range $[A_e,B_e]$ within which it may be reassigned; the kernel then has $7k$ vertices. Finally, Tree Metric Violation Distance is fixed-parameter tractable and solvable in $k^{O(k)}n^{O(1)}$ time.

Problem

• Metric Violation Distance
• Parameterized Complexity
• Kernelization
• Tree Metrics
• Noisy Data Repair

Innovation

• Metric Violation Distance
• Linear Kernel
• Single-Exponential Algorithm
• Fixed-Parameter Tractability
• Tree Metrics