In the Categorical Clustering problem, we are given a set of vectors (matrix) A={a_1,\ldots,a_n} over Σ^m, where Σis a finite alphabet, and integers k and B. The task is to partition A into k clusters such that the median objective of the clustering in the Hamming norm is at most B. That is, we seek a partition {I_1,\ldots,I_k} of {1,\ldots,n} and vectors c_1,\ldots,c_k\inΣ^m such that \sum_{i=1}^k\sum_{j\in I_i}d_h(c_i,a_j)\leq B, where d_H(a,b) is the Hamming distance between vectors a and b. Fomin, Golovach, and Panolan [ICALP 2018] proved that the problem is fixed-parameter tractable (for binary case Σ={0,1}) by giving an algorithm that solves the problem in time 2^{O(B\log B)} (mn)^{O(1)}. We extend this algorithmic result to a popular capacitated clustering model, where in addition the sizes of the clusters should satisfy certain constraints. More precisely, in Capacitated Clustering, in addition, we are given two non-negative integers p and q, and seek a clustering with p\leq |I_i|\leq q for all i\in{1,\ldots,k}. Our main theorem is that Capacitated Clustering is solvable in time 2^{O(B\log B)}|Σ|^B(mn)^{O(1)}. The theorem not only extends the previous algorithmic results to a significantly more general model, it also implies algorithms for several other variants of Categorical Clustering with constraints on cluster sizes.
Clustering under Constraints: Efficient Parameterized Approximation Schemes
Indian Institute of Technology Bombay（孟买印度理工学院）；CISPA Helmholtz Center for Information Security（CISPA赫尔姆霍兹信息安全中心）；Indian Institute of Technology Jodhpur（焦特普尔印度理工学院）
Parameterized Approximation Schemes for Clustering with General Norm Objectives
University of Wrocław（弗罗茨瓦夫大学）；Aalto University（阿尔托大学）；University of British Columbia（不列颠哥伦比亚大学）；CISPA Helmholtz Center for Information Security（CISPA亥姆霍兹信息安全中心）；Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）；The University of Sheffield（谢菲尔德大学）
Polynomial Time Approximation Schemes for Clustering in Low Highway Dimension Graphs
我们研究聚类问题，如k-Median，k-Means，和设施位置的低维公路图，这是一个图形参数建模的交通网络。之前已经证明了这些问题的近似方案存在，它们可以在准多项式时间内运行（假设高速公路维度恒定）[Feldmann et al. SICOMP 2018]，也可以在FPT时间内运行（通过聚类数$k$、高速公路维度和近似因子进行参数化）[Becker et al. ESA~2018，Braverman et al. 2020]。在本文中，我们表明，多项式时间近似方案（PTAS）存在（假设恒定的公路尺寸）。我们还表明，所考虑的问题是NP-困难的公路维数为1的图。
