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We consider the problem of automating the derivation of tight asymptotic complexity bounds for solving Horn clauses. Clearly, the solving time crucially depends on the "sparseness" of the computed relations. Therefore, our asymptotic runtime analysis is accompanied by an asymptotic sparsity calculus together with an asymptotic sparsity analysis. The technical problem here is that least fixpoint iteration fails on asymptotic complexity expressions: the intuitive reason is that O(1)+ O(1) = O(1) but O(1) + ... + O(1) may return any value.