Clustering to Given Connectivities
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We define a general variant of the graph clustering problem where the criterion of density for the clusters is (high) connectivity. In Clustering to Given Connectivities, we are given an n-vertex graph G, an integer k, and a sequence Λ=〈λ_1,...,λ_t〉 of positive integers and we ask whether it is possible to remove at most k edges from G such that the resulting connected components are exactly t and their corresponding connectivities are lower-bounded by the numbers in Λ. We prove that this problem, parameterized by k, is fixed parameter tractable i.e., can be solved by an f(k)· n^O(1)-step algorithm, for some function f that depends only on the parameter k. Our algorithm uses the recursive understanding technique that is especially adapted so to deal with the fact that, in out setting, we do not impose any restriction to the connectivity demands in Λ. We also prove that Clustering to Given Connectivities, parameterized by k, does not admit a polynomial kernel unless NP⊆co-NP/poly.
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