We revisit the capacitated $k$-median problem, focusing on constant-factor approximation algorithms with either $(1+ε)$-cardinality or capacity violations. Previous results [DemirciL16, Li16} achieve an $O_ε(1)$-approximation with $(1+ε)$-violations via complex extended configuration linear programming formulations with the round-or-cut framework. We propose a simpler, unified round-or-cut framework that handles both cardinality and capacity violation settings. Our main technical contribution is a mechanism that directly utilizes the costs of local instances to generate valid inequalities. Specifically, we partition the metric space into local pieces of size $O(1/ε)$ as in previous work, and construct instances of capacitated clustering with penalties, which allow client demands to be unserved at a penalty cost. Solving these local instances yields lower bounds on the original LP cost, yielding valid LP constraints. In contrast, previous approaches rely on complex extended configuration LPs over connection variables $x_{ij}$'s and opening variables $y_i$'s. As a secondary contribution, our approach achieves an improved running time of $\text{poly}(n, 1/ε)$, bypassing the exponential dependence on $1/ε$ required to solve the previous extended configuration LPs. Instead of solving the local instances optimally in time exponential in $1/ε$, we obtain $O(1/ε)$-approximate solutions in $\text{poly}(n, 1/ε)$ time. Finally, for the cardinality-violation setting, our algorithm opens a set of at most $(1+ε)k$ $\textit{distinct}$ facilities, and thus gives a bicriteria approximation in the $\textit{hard-capacitated}$ setting. In contrast, the algorithm of [Li16] may open the same facility multiple times.

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