{"id":964224,"date":"2023-08-27T12:28:52","date_gmt":"2023-08-27T19:28:52","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=964224"},"modified":"2023-10-04T15:29:46","modified_gmt":"2023-10-04T22:29:46","slug":"improved-diversity-maximization-algorithms-for-matching-and-pseudoforest","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/improved-diversity-maximization-algorithms-for-matching-and-pseudoforest\/","title":{"rendered":"Improved Diversity Maximization Algorithms for Matching and Pseudoforest"},"content":{"rendered":"

In this work we consider the diversity maximization problem, where given a data set X<\/span><\/span><\/span><\/span> of n<\/span><\/span><\/span><\/span> elements, and a parameter k<\/span><\/span><\/span><\/span>, the goal is to pick a subset of X<\/span><\/span><\/span><\/span> of size k<\/span><\/span><\/span><\/span> maximizing a certain diversity measure. [CH01] defined a variety of diversity measures based on pairwise distances between the points. A constant factor approximation algorithm was known for all those diversity measures except “remote-matching”, where only an O<\/span>(<\/span>log <\/span><\/span>k<\/span>)<\/span><\/span><\/span><\/span> approximation was known. In this work we present an O<\/span>(<\/span>1<\/span>)<\/span><\/span><\/span><\/span> approximation for this remaining notion. Further, we consider these notions from the perspective of composable coresets. [IMMM14] provided composable coresets with a constant factor approximation for all but “remote-pseudoforest” and “remote-matching”, which again they only obtained a O<\/span>(<\/span>log <\/span><\/span>k<\/span>)<\/span><\/span><\/span><\/span> approximation. Here we also close the gap up to constants and present a constant factor composable coreset algorithm for these two notions. For remote-matching, our coreset has size only O<\/span>(<\/span>k<\/span>)<\/span><\/span><\/span><\/span>, and for remote-pseudoforest, our coreset has size O<\/span>(<\/span>k^{<\/span>1<\/span>+<\/span>\u03b5}<\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span> for any \u03b5<\/span>><\/span>0<\/span><\/span><\/span><\/span>, for an O<\/span>(<\/span>1<\/span>\/<\/span><\/span><\/span>\u03b5<\/span>)<\/span><\/span><\/span><\/span>-approximate coreset.<\/p>\n","protected":false},"excerpt":{"rendered":"

In this work we consider the diversity maximization problem, where given a data set X of n elements, and a parameter k, the goal is to pick a subset of X of size k maximizing a certain diversity measure. [CH01] defined a variety of diversity measures based on pairwise distances between the points. A constant 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