Max-sum diversity via convex programming

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(·) that measures diversity of a subset, the task is to select a feasible subset S such that f(S) is maximized. The sum-dispersion function f(S) = sumx,y ∈ 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 max-sum or 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(·,·) of negative type. Distances of negative type are, for example, metric distances stemming from the ℓ2 and ℓ1 norm, as well as the cosine or spherical, or Jaccard distance which are popular similarity metrics in web and image search.