Max-Sum diversification, monotone submodular functions and dynamic updates

Abstract

Result diversification has many important applications in databases, operations research, information retrieval, and finance. In this paper, we study and extend a particular version of result diversification, known as max-sum diversification. More specifically, we consider the setting where we are given a set of elements in a metric space and a set valuation function f defined on every subset. For any given subset S, the overall objective is a linear combination of f(S) and the sum of the distances induced by S. The goal is to find a subset S satisfying some constraints that maximizes the overall objective. This problem is first studied by Gollapudi and Sharma in [17] for modular set functions and for sets satisfying a cardinality constraint (uniform matroids). In their paper, they give a 2-approximation algorithm by reducing to an earlier result in [20]. The first part of this paper considers an extension of the modular case to the monotone submodular case, for which the algorithm in [17] no longer applies. Interestingly, we are able to maintain the same 2-approximation using a natural, but different greedy algorithm. We then further extend the problem by considering any matroid constraint and show that a natural single swap local search algorithm provides a 2-approximation in this more general setting. This extends the Nemhauser, Wolsey and Fisher approximation result [20] for the problem of submodular function maximization subject to a matroid constraint (without the distance function component). The second part of the paper focuses on dynamic updates for the modular case. Suppose we have a good initial approximate solution and then there is a single weight-perturbation either on the valuation of an element or on the distance between two elements. Given that users expect some stability in the results they see, we ask how easy is it to maintain a good approximation without significantly changing the initial set. We measure this by the number of updates, where each update is a swap of a single element in the current solution with a single element outside the current solution. We show that we can maintain an approximation ratio of 3 by just a single update if the perturbation is not too large.