Local Search for Max-Sum Diversification

TL;DR

This paper introduces efficient local search algorithms with approximation guarantees for max-sum diversification problems, including those with matroid constraints, significantly improving scalability and applicability in diverse settings.

Contribution

It presents new polynomial-time approximation schemes for max-sum diversification, extending to matroid constraints and combining with submodular functions, using simple local search techniques.

Findings

Achieves a (1-O(1/k))-approximation for negative type distances with matroid constraints.

Provides a PTAS with linear dependence on data size, outperforming previous convex optimization methods.

Extends techniques to optimize combined diversity and relevance objectives with near-optimal guarantees.

Abstract

We provide simple and fast polynomial time approximation schemes (PTASs) for several variants of the max-sum diversification problem which, in its most basic form, is as follows: Given n points p_1,...,p_n in R^d and an integer k, select k points such that the average Euclidean distance between these points is maximized. This problem commonly appears in information retrieval and web-search in order to select a diverse set of points from the input. In this context, it has recently received a lot of attention. We present new techniques to analyze natural local search algorithms. This leads to a (1-O(1/k))-approximation for distances of negative type, even subject to any matroid constraint of rank k, in time O(n k^2 log k), when assuming that distance evaluations and calls to the independence oracle are constant time. Negative type distances include as special cases Euclidean distances and many further natural distances. Our result easily transforms into a PTAS and improves on the only previously known PTAS for this setting, which relies on convex optimization techniques in an n-dimensional space and is impractical for large data sets. In contrast, our procedure has an (optimal) linear dependence on n. Using generalized exchange properties of matroid intersection, we show that a PTAS can be obtained for matroid intersection constraints as well. Moreover, our techniques, being based on local search, are conceptually simple and allow for various extensions. In particular, we get asymptotically optimal O(1)-approximations when combining the classic dispersion function with a monotone submodular objective, which is a very common class of functions to measure diversity and relevance. This result leverages recent advances on local search techniques based on proxy functions to obtain optimal approximations for monotone submodular function maximization subject to a matroid constraint.