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Constraint-based mining techniques on sequence databases have been studied extensively the last few years and efficient algorithms enable to compute complete collections of patterns (e.g., sequences) which satisfy conjunctions of monotonic and/or anti-monotonic constraints. Studying new applications of these techniques, we believe that a primitive constraint which enforces enough similarity w.r.t a given reference sequence would be extremely useful and should benefit from such a recent algorithmic breakthrough. A non trivial similarity constraint is however neither monotonic nor anti-monotonic. Therefore, we have studied its definition as a conjunction of two constraints which satisfy the desired monotonicity properties: a pattern is called similar to a reference pattern x when its longest common subsequence with x (LCS) is large enough (i.e., a monotonic part) and when the number of deletions such that it becomes the LCS is small enough (i.e., an anti-monotonic part). We provide an experimental validation which confirms the added value of this approach on a biological database. Classical issues like scalability and pruning efficiency are discussed.