Parameterized Complexity of Categorical Clustering with Size Constraints

TL;DR

This paper extends fixed-parameter tractable algorithms for categorical clustering with Hamming distance to include size constraints on clusters, broadening applicability to more complex clustering scenarios.

Contribution

It introduces an algorithm for capacitated categorical clustering, incorporating cluster size constraints, building on prior work for the binary case.

Findings

Algorithm solves capacitated clustering in 2^{O(B log B)} ||^B (mn)^{O(1)} time.

Extends previous algorithms to more general clustering models.

Enables solutions for various constrained clustering variants.

Abstract

In the Categorical Clustering problem, we are given a set of vectors (matrix) A={a_1,\ldots,a_n} over \Sigma^m, where \Sigma is 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\in\Sigma^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 \Sigma={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)}|\Sigma|^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.