Max-sum diversity via convex programming

TL;DR

This paper introduces a polynomial-time approximation scheme for maximizing sum-sum diversity under matroid constraints for negative type distances, applicable to various popular similarity metrics.

Contribution

It provides the first PTAS for max-sum diversification with negative type distances under matroid constraints, expanding the scope of efficient algorithms in diversity maximization.

Findings

Developed a PTAS for max-sum diversification with negative type distances.

Applicable to common similarity metrics like Euclidean, cosine, and Jaccard.

Enhances algorithmic tools for diversity maximization in information retrieval and related fields.

Abstract

Diversity maximization is an important concept in information retrieval, computational geometry and operations research. Usually, it is a variant of the following problem: Given a ground set, constraints, and a function $f(\cdot)$ that measures diversity of a subset, the task is to select a feasible subset $S$ such that $f(S)$ is maximized. The \emph{sum-dispersion} function $f(S) = \sum_{x,y \in S} d(x,y)$, which is the sum of the pairwise distances in $S$, is in this context a prominent diversification measure. The corresponding diversity maximization is the \emph{max-sum} or \emph{sum-sum diversification}. Many recent results deal with the design of constant-factor approximation algorithms of diversification problems involving sum-dispersion function under a matroid constraint. In this paper, we present a PTAS for the max-sum diversification problem under a matroid constraint for distances $d(\cdot,\cdot)$ of \emph{negative type}. Distances of negative type are, for example, metric distances stemming from the $\ell_2$ and $\ell_1$ norm, as well as the cosine or spherical, or Jaccard distance which are popular similarity metrics in web and image search.