Chasing Submodular Objectives, and Submodular Maximization via Cutting Planes

2025-11-17

AI Summary

This paper studies the dynamic submodular objective chasing problem: a sequence of constrained (cardinality- or partition-matroid-constrained) submodular maximization problems is revealed over time, with both objective functions and feasible domains evolving dynamically. The goal is to approximate the offline optimal solution while minimizing the total number of adjustments (i.e., solution changes). To this end, we propose the first “approximate-or-separate” meta-algorithm that integrates cutting-plane methods with the continuous greedy framework—marking the first application of cutting-plane techniques to constrained submodular maximization. Leveraging the multilinear extension, improved rounding schemes, and a separation oracle, our algorithm achieves a $(1-1/e-varepsilon)$-approximation ratio and optimal competitive ratio in adjustment cost. Our results unify and extend to downstream applications including static submodular optimization and communication complexity.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We introduce the emph{submodular objectives chasing problem}, which generalizes many natural and previously-studied problems: a sequence of constrained submodular maximization problems is revealed over time, with both the objective and available ground set changing at each step. The goal is to maintain solutions of high approximation and low total emph{recourse} (number of changes), compared with exact offline algorithms for the same input sequence. For the central cardinality constraint and partition matroid constraints we provide polynomial-time algorithms achieving both optimal $(1-1/e-ε)$-approximation and optimal competitive recourse for emph{any} constant-approximation. Key to our algorithm's polynomial time, and of possible independent interest, is a new meta-algorithm for $(1-1/e-ε)$-approximately maximizing the multilinear extension under general constraints, which we call {em approximate-or-separate}. Our algorithm relies on an improvement of the round-and-separate method [Gupta-Levin SODA'20], inspired by an earlier proof by [Vondrák, PhD~Thesis'07]. The algorithm, whose guarantees are similar to the influential {em continuous greedy} algorithm [Calinescu-Chekuri-Pál-Vondrák SICOMP'11], can use any cutting plane method and separation oracle for the constraints. This allows us to introduce cutting plane methods, used for exact unconstrained submodular minimization since the '80s [Grötschel/Lovász/Schrijver Combinatorica'81], as a useful method for (optimal approximate) constrained submodular maximization. We show further applications of this approach to static algorithms with curvature-sensitive approximation, and to communication complexity protocols.

Problem

• Solving sequential submodular maximization with changing objectives and constraints
• Achieving optimal approximation with minimal solution changes over time
• Developing cutting plane methods for constrained submodular optimization problems

Innovation

• Approximate-or-separate method for multilinear extension
• Cutting plane techniques for constrained submodular maximization
• Improved round-and-separate approach with polynomial runtime