Max-Sum Diversification, Monotone Submodular Functions and Semi-metric Spaces

TL;DR

This paper investigates the max-sum diversification problem in semi-metric spaces, relaxing the triangle inequality to analyze how it affects approximation ratios for selecting diverse, high-quality subsets under matroid constraints.

Contribution

It extends prior work by relaxing the triangle inequality assumption, providing new bounds on approximation ratios for diversification problems in semi-metric spaces.

Findings

Approximation ratios depend on the relaxation parameter of the triangle inequality.

Results apply to both uniform and arbitrary matroid constraints.

Theoretical bounds improve understanding of diversification in semi-metric spaces.

Abstract

In many applications such as web-based search, document summarization, facility location and other applications, the results are preferable to be both representative and diversified subsets of documents. The goal of this study is to select a good "quality", bounded-size subset of a given set of items, while maintaining their diversity relative to a semi-metric distance function. This problem was first studied by Borodin et al\cite{borodin}, but a crucial property used throughout their proof is the triangle inequality. In this modified proof, we want to relax the triangle inequality and relate the approximation ratio of max-sum diversification problem to the parameter of the relaxed triangle inequality in the normal form of the problem (i.e., a uniform matroid) and also in an arbitrary matroid.