Ranking with Diverse Intents and Correlated Contents

TL;DR

This paper introduces an approximation algorithm for a document ranking problem that accounts for diverse user interests and correlated content, generalizing previous models and extending to complex valuation functions.

Contribution

The paper presents an O(ρ)-approximation algorithm for ranking with diverse intents and correlated contents, extending previous models to multiple interest sets and XOS valuations.

Findings

Achieves an O(ρ)-approximation where ρ is the integrality gap of the set cover LP

Generalizes constant approximations for min-sum set cover and related problems

Extends results to multiple interest sets and XOS valuation functions

Abstract

We consider the following document ranking problem: We have a collection of documents, each containing some topics (e.g. sports, politics, economics). We also have a set of users with diverse interests. Assume that user u is interested in a subset I_u of topics. Each user u is also associated with a positive integer K_u, which indicates that u can be satisfied by any K_u topics in I_u. Each document s contains information for a subset C_s of topics. The objective is to pick one document at a time such that the average satisfying time is minimized, where a user's satisfying time is the first time that at least K_u topics in I_u are covered in the documents selected so far. Our main result is an O({\rho})-approximation algorithm for the problem, where {\rho} is the algorithmic integrality gap of the linear programming relaxation of the set cover instance defined by the documents and topics. This result generalizes the constant approximations for generalized min-sum set cover and ranking with unrelated intents and the logarithmic approximation for the problem of ranking with submodular valuations (when the submodular function is the coverage function), and can be seen as an interpolation between these results. We further extend our model to the case when each user may interest in more than one sets of topics and when the user's valuation function is XOS, and obtain similar results for these models.