An Asymptotically Optimal Approximation Algorithm for Multiobjective Submodular Maximization at Scale

TL;DR

This work tackles multiobjective submodular maximization under a cardinality constraint, seeking a single set that nearly optimizes the minimum across many submodular objectives. It introduces a discrete, scalable algorithm that achieves the near-optimal approximation $1-1/e-\varepsilon$ without relying on the expensive multilinear extension, using a carefully designed LP-based sampling step and a concentration-based analysis. A key novelty is the pre-processing step and the use of multiplicative weights updates to solve the LP efficiently and enable lazy evaluations, yielding practical running times. The paper further applies the framework to fair centrality maximization and demonstrates superior performance on large-scale experiments, underscoring the method’s applicability to fairness-aware network design tasks with real-world impact.

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, 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.

Figures (29)

• Figure 1: Multiobjective submodular maximization for max-$k$-cover. We use $k=20$ Kronecker graphs on $n=64$ nodes. We show the function value (left) and the number of evaluations (right). We report mean and standard deviation over 5 random instances.
• Figure 2: Fair centrality maximization on the Amazon co-purchasing graph Arts, Crafts & Sewing with $n=5051$ nodes and $k=2$ colors.
• Figure 3: Fair influence maximization on a simulated Antelope Valley network of $n\!=\!500$ nodes on attribute ethnicity with $k\!=\!5$.
• Figure 4: We run Algorithm \ref{['alg:multiobjective']} on a max-$k$-cover instance of $k=20$ Erdős-Rényi random graphs on $n=64$ nodes with $p=0.1$. For the left plot, we vary the number of repetitions while using $\phi=10$. For the right plot, we vary the factor $\phi$ while using $20$ repetitions. We report mean and standard deviation over 5 runs.
• Figure 5: Multiobjective submodular maximization for max-$k$-cover. We use $k=20$ Barabási-Albert graphs on $n=64$ nodes. We show the function value (top) and the number of evaluations (bottom). We report mean and standard deviation over 5 runs.

Theorems & Definitions (28)

• Lemma 4.1
• proof
• Theorem 4.2
• Lemma 4.2
• Lemma 4.2
• Theorem 4.3: Corollary 16 in Kuszmaul+21
• Lemma 4.3
• proof
• proof : Proof of Theorem \ref{['thm:multiobjective']}
• Lemma 4.3