Parameterized k-Clustering: The distance matters!
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We consider the k-Clustering problem, which is for a given multiset of n vectors X⊂Z^d and a nonnegative number D, to decide whether X can be partitioned into k clusters C_1, ..., C_k such that the cost ∑_i=1^k _c_i∈R^d∑_x ∈ C_ix-c_i_p^p ≤ D, where ·_p is the Minkowski (L_p) norm of order p. For p=1, k-Clustering is the well-known k-Median. For p=2, the case of the Euclidean distance, k-Clustering is k-Means. We show that the parameterized complexity of k-Clustering strongly depends on the distance order p. In particular, we prove that for every p∈ (0,1], k-Clustering is solvable in time 2^O(D D) (nd)^O(1), and hence is fixed-parameter tractable when parameterized by D. On the other hand, we prove that for distances of orders p=0 and p=∞, no such algorithm exists, unless FPT=W[1].
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