Submodular Maximization Subject to Uniform and Partition Matroids: From Theory to Practical Applications and Distributed Solutions

TL;DR

The paper surveys submodular maximization under uniform and partition matroid constraints, addressing the NP-hard nature of the problem while detailing polynomial-time, approximation-guaranteed methods. It contrasts discrete sequential greedy and continuous relaxation via the multilinear extension and matroid polytope, highlighting guarantees such as $f(\mathcal{S}_{SG}) \ge (1-1/e)OPT$ for the continuous approach and curvature-dependent bounds $\alpha_{\text{uniform}}=\frac{1}{c}\left(1-\frac{1}{e^{c}}\right)$ and $\alpha_{\text{partition}}=\frac{1}{1+c}$ for greedy under uniform and partition matroids, respectively. The work further extends these ideas to distributed settings, addressing large-scale data, privacy, and communication challenges, and connects theory to diverse applications like exemplar clustering, sensor placement, information harvesting, traffic identifiability, and persistent monitoring. It also discusses practical issues such as gradient estimation, information-graph connectivity, and probabilistic message-passing schemes, outlining a roadmap for future research in deep submodular functions, online variants, and fairness considerations. Overall, the paper provides a cohesive framework bridging foundational theory and practical, scalable optimization in networked and distributed environments.

Abstract

This article provides a comprehensive exploration of submodular maximization problems, focusing on those subject to uniform and partition matroids. Crucial for a wide array of applications in fields ranging from computer science to systems engineering, submodular maximization entails selecting elements from a discrete set to optimize a submodular utility function under certain constraints. We explore the foundational aspects of submodular functions and matroids, outlining their core properties and illustrating their application through various optimization scenarios. Central to our exposition is the discussion on algorithmic strategies, particularly the sequential greedy algorithm and its efficacy under matroid constraints. Additionally, we extend our analysis to distributed submodular maximization, highlighting the challenges and solutions for large-scale, distributed optimization problems. This work aims to succinctly bridge the gap between theoretical insights and practical applications in submodular maximization, providing a solid foundation for researchers navigating this intricate domain.

Figures (4)

• Figure 1: Properties of submodular functions.
• Figure 2: Multilinear extension extends a submodular function $f(\mathcal{R})$ to the continuous space defined on hypercube $[0,1]^n$. $F(\boldsymbol{x})$ agrees with $f(\mathcal{R})$ on the vertices of the hypercube (for integral $\boldsymbol{x}$).
• Figure 3: A simple schematic illustration of a distributed sensor deployment for data harvesting involving a group of agents $\mathcal{A}$. Each agent $i \in \mathcal{A}$ possesses $\kappa_i$ data harvesting drones, which can be deployed at a set of pre-assigned deployment points $\mathcal{B}_i$, indicated by the same color used to depict the agent. The agents communicate over a connected graph to determine the optimal deployment positions for the team, as defined by the submodular maximization problem subject to a partition matroid described in Section \ref{['sec::applications']}.
• Figure 4: The top figures illustrate the communication topology (solid black arrows) and the message-passing sequence (dashed red arrows). The bottom figures detail the information-sharing graph. In case (a), the communication graph possesses a Hamiltonian path, enabling the shortest sequence for optimal message passing in the sequential greedy algorithm. In case (b), the absence of a Hamiltonian path in the communication graph requires the blue agent to participate twice in the message-passing process to implement the sequential greedy algorithm. In case (c), with the communication graph disconnected, the information-sharing graph becomes incomplete, preventing the precise execution of the sequential greedy algorithm. Consequently, the guaranteed optimality gap decreases from $1/2$.

Theorems & Definitions (7)

• Definition 1: Normal set function
• Definition 2: Monotone increasing set function
• Definition 3: Submodular function
• Definition 4: Modular set function
• Definition 5
• Definition 6: Uniform Matroid
• Definition 7: Partition Matroid