In this work, we study the $k$-median clustering problem with an additional equal-size constraint on the clusters, from the perspective of parameterized preprocessing. Our main result is the first lossy ($2$-approximate) polynomial kernel for this problem, parameterized by the cost of clustering. We complement this result by establishing lower bounds for the problem that eliminate the existences of an (exact) kernel of polynomial size and a PTAS.

翻译：在这项工作中,我们从参数化预处理的角度来看,研究美元中位群集问题,对集群附加同等规模的限制,我们的主要结果就是这一问题的第一次损失(大约为2美元)多面内核,按聚集成本参数计算。我们通过为消除多面体大小(具体)内核和PTAS存在的问题确定较低的界限来补充这一结果。