Max-Sum Diversification, Monotone Submodular Functions and Dynamic Updates

Allan Borodin; Aadhar Jain; Hyun Chul Lee; Yuli Ye

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arXiv · 1203.6397

Max-Sum Diversification, Monotone Submodular Functions and Dynamic Updates

Allan Borodin·Aadhar Jain·Hyun Chul Lee·Yuli Ye

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Abstract

Result diversification is an important aspect in web-based search, document summarization, facility location, portfolio management and other applications. Given a set of ranked results for a set of objects (e.g. web documents, facilities, etc.) with a distance between any pair, the goal is to select a subset $S$ satisfying the following three criteria: (a) the subset $S$ satisfies some constraint (e.g. bounded cardinality); (b) the subset contains results of high "quality"; and (c) the subset contains results that are "diverse" relative to the distance measure. The goal of result diversification is to produce a diversified subset while maintaining high quality as much as possible. We study a broad class of problems where the distances are a metric, where the constraint is given by independence in a matroid, where quality is determined by a monotone submodular function, and diversity is defined as the sum of distances between objects in $S$. Our problem is a generalization of the {\em max sum diversification} problem studied in \cite{GoSh09} which in turn is a generaliztion of the {\em max sum $p$-dispersion problem} studied extensively in location theory. It is NP-hard even with the triangle inequality. We propose two simple and natural algorithms: a greedy algorithm for a cardinality constraint and a local search algorithm for an arbitary matroid constraint. We prove that both algorithms achieve constant approximation ratios.

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BibTeX RIS

Allan Borodin, Aadhar Jain, Hyun Chul Lee, Yuli Ye. 2012-03-28. Max-Sum Diversification, Monotone Submodular Functions and Dynamic Updates. https://arxiv.org/abs/1203.6397

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