AI 中文总结
本文针对无权图和加权图的最小一致子集问题，构建细粒度复杂性图谱，提出加权图MCS的改进算法并补充下界，改进无权MCS算法，界定一致子集选择的算法边界。
AI 中文摘要
实例选择是缓解大规模监督聚类中最近邻分类计算瓶颈的关键技术，该目标的经典理论形式化为最小一致子集（Minimum Consistent Subset，MCS）问题。近期研究已探索其在无权图上的复杂性以揭示可处理性的结构边界，但任意度量空间更适合用（边）加权图精确建模。本文针对无权图和加权图上的MCS，构建了全面的细粒度复杂性图谱。核心成果为：针对树宽为tw、含c种颜色的n顶点加权图MCS实例，提出了3^{c·(tw+1)}·n^{tw+O(1)}算法，在通用性和运行时间上均显著优于此前针对树结构无权MCS的最优算法。同时，补充了一系列下界结果，在指数时间假设（Exponential Time Hypothesis，ETH）下排除了加权图和无权图运行时间的渐近改进可能；此外，将近期基于顶点覆盖的无权MCS的略超指数算法（AAAI 2026）改进为单指数算法，并在ETH下排除了其进一步改进为亚指数运行时间的可能。综上，本文结果严格界定了不同度量结构下一致子集选择的算法边界。
英文摘要
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.
