Improved Learning with Structure: Fine-Grained Complexity of Minimum Consistent Subset

2026-07-30

AI Summary

This work investigates the fine-grained complexity of the Minimum Consistent Subset (MCS) problem for nearest neighbor classification on both weighted and unweighted graphs. By leveraging treewidth as a structural parameter, the study presents the first efficient algorithm for weighted graphs with running time $3^{c\cdot(\text{tw}+1)} \cdot n^{\text{tw}+O(1)}$, substantially improving upon prior results and also refining the single-exponential algorithm for unweighted graphs. Under the Exponential Time Hypothesis (ETH), the authors establish the tightness of this time bound, demonstrating that further asymptotic speedup is unlikely. Combining treewidth-based dynamic programming, fine-grained complexity analysis, and vertex cover techniques, the paper provides a comprehensive characterization of the computational limits of MCS across diverse graph classes.

Abstract

Instance selection is a vital technique for mitigating the computational bottlenecks of nearest-neighbor classification in large-scale supervised clustering. A classical theoretical formulation of this objective is the Minimum Consistent Subset (MCS) problem. While recent research has explored its complexity on unweighted graphs to uncover structural boundaries of tractability, arbitrary metric spaces are much more accurately modeled by (edge-)weighted graphs. In this paper, we develop a comprehensive fine-grained complexity map of MCS on both unweighted and weighted graphs. As our main result, we introduce a $3^{c \cdot(\mathrm{tw}+1)}\cdot n^{\mathrm{tw}+\mathcal{O}(1)}$ algorithm for $n$-vertex $c$-colored MCS instances on weighted graphs of treewidth $\mathrm{tw}$, substantially improving upon the previous state-of-the-art algorithm for unweighted MCS on trees both in terms of generality and running time. We complement this positive result with a series of lower bounds that rule out asymptotic improvements to the running time for both weighted and unweighted graphs under the Exponential Time Hypothesis (ETH). Moreover, we improve the recent slightly superexponential vertex-cover based algorithm for unweighted MCS (AAAI 2026) to a single-exponential one, and rule out further improvements to subexponential running times under the ETH. Together, our results strictly delineate the algorithmic boundaries of consistent subset selection across diverse metric structures.

Problem

• Minimum Consistent Subset
• fine-grained complexity
• weighted graphs
• treewidth
• instance selection

Innovation

• Minimum Consistent Subset
• fine-grained complexity
• treewidth
• weighted graphs
• Exponential Time Hypothesis