An Asymptotically Optimal Approximation Algorithm for Multiobjective Submodular Maximization at Scale

Fabian Christian Spaeh, Atsushi Miyauchi

TL;DR: We provide a new and efficient approximation algorithm for multiobjective submodular maximization

Abstract: Maximizing a single submodular set function subject to a cardinality constraint is a well-studied and central topic in combinatorial optimization. However, finding a set that maximizes multiple functions at the same time is much less understood, even though it is a formulation which naturally occurs in robust maximization or problems with fairness considerations such as fair influence maximization or fair allocation. In this work, we consider the problem of maximizing the minimum over many submodular functions subject to a cardinality constraint, which is known as multiobjective submodular maximization. All known polynomial-time approximation algorithms either obtain a weak approximation guarantee or rely on the evaluation of the multilinear extension. The latter is expensive to evaluate and renders such algorithms impractical. We bridge this gap and introduce the first scalable and practical algorithm that obtains the best-known approximation guarantee. We furthermore introduce a novel application fair centrality maximization and show how it can be addressed via multiobjective submodular maximization. In our experimental evaluation, we show that our algorithm outperforms known algorithms in terms of objective value and running time.

Lay Summary:

Many decision-making problems in machine learning and AI involve selecting a limited set of items to optimize multiple objectives at once. For instance, when ensuring fairness across different groups. This is challenging when these objectives conflict and the mathematical tools used in single-objective problems no longer apply. Previous algorithms that give strong guarantees often rely on complex continuous relaxations that are too slow for large datasets, while faster methods offer much weaker performance.

We designed a new algorithm that directly tackles this multiobjective optimization problem in a scalable and principled way. Our method avoids the need for costly continuous relaxations and instead uses a novel probabilistic strategy with strong theoretical backing. It nearly matches the best-known performance guarantee and works efficiently even on large datasets.

To demonstrate its usefulness, we applied our algorithm to a new fairness-oriented problem: ensuring that a target node in a network is visible to all demographic groups. We also tested it on problems like fair influence and coverage maximization, where it consistently outperformed previous methods in both accuracy and speed.

Link To Code: https://github.com/285714/multiobjective

Primary Area: Optimization->Discrete and Combinatorial Optimization

Keywords: submodular maximization, multiobjective, max-min fairness

Submission Number: 8187