Max-Sum Diversification, Monotone Submodular Functions and Dynamic Updates
This addresses result diversification in applications like web search and facility location, offering incremental improvements to existing methods.
The paper tackles the problem of result diversification by selecting a subset that satisfies constraints, maintains high quality via a monotone submodular function, and maximizes diversity as the sum of distances, proposing greedy and local search algorithms with constant approximation ratios for NP-hard cases.
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.