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Finding the Jaccard median

The median problem in the weighted Jaccard metric was analyzed by Späth in 1981. Up until now, only an exponential-time exact algorithm was known. We (a) obtain a PTAS for the weighted Jaccard median problem and (b) show that the problem does not admit a FPTAS (assuming P ≠ NP), even when restricted to binary vectors. The PTAS is built on a number of different algorithmic ideas and the hardness result makes use of an especially interesting gadget.