Ranking with submodular functions on a budget

TL;DR

This paper introduces a novel ranking problem based on submodular functions with budget constraints, proposing algorithms with approximation guarantees and demonstrating their effectiveness through empirical evaluation.

Contribution

It formulates the max-submodular ranking problem, develops practical algorithms with approximation guarantees, and empirically shows their superior performance over baselines.

Findings

Proposed algorithms achieve near-optimal solutions under budget constraints.

Empirical results show significant improvement over baseline methods.

Algorithms are effective for both cardinality and knapsack constraints.

Abstract

Submodular maximization has been the backbone of many important machine-learning problems, and has applications to viral marketing, diversification, sensor placement, and more. However, the study of maximizing submodular functions has mainly been restricted in the context of selecting a set of items. On the other hand, many real-world applications require a solution that is a ranking over a set of items. The problem of ranking in the context of submodular function maximization has been considered before, but to a much lesser extent than item-selection formulations. In this paper, we explore a novel formulation for ranking items with submodular valuations and budget constraints. We refer to this problem as max-submodular ranking (MSR). In more detail, given a set of items and a set of non-decreasing submodular functions, where each function is associated with a budget, we aim to find a ranking of the set of items that maximizes the sum of values achieved by all functions under the budget constraints. For the MSR problem with cardinality- and knapsack-type budget constraints we propose practical algorithms with approximation guarantees. In addition, we perform an empirical evaluation, which demonstrates the superior performance of the proposed algorithms against strong baselines.