Parameterized complexity of categorical clustering with size constraints

In the Categorical Clustering problem, we are given a set of vectors (matrix) A={a1,…,an} over Σm, where Σ is finite alphabet, and integers k B. The task to partition A into clusters such that median objective clustering in Hamming norm at most Fomin, Golovach, Panolan [ICALP 2018] proved problem fixed-parameter tractable for binary case Σ={0,1}. We extend this algorithmic result popular capacitated model, addition sizes lower upper bounded by integer parameters p q, respectively. Our main theorem solvable time 2O(Blog⁡B)|Σ|B⋅(mn)O(1).