Approximating Submodular Matroid-Constrained Partitioning

2025-06-24

AI Summary

This paper studies the submodular partition problem under matroid constraints: minimizing the sum of a submodular function over disjoint subsets, subject to the constraint that the partition separates some basis of a given matroid. This unified framework captures classical problems including multiway cut, hypergraph $k$-cut, and matrix multicut. We introduce and systematically analyze this general model for the first time. For symmetric, monotone, and general submodular functions, we achieve the current best approximation ratios: $(2 - 2/k)$—matching the theoretical lower bound for symmetric functions—and the first nontrivial efficient approximations for monotone and general cases. Our approach integrates submodular optimization, matroid theory, and combinatorial approximation algorithms, combining Lagrangian relaxation with greedy construction techniques. The results advance the unified modeling and tight approximation boundary analysis of submodular partitioning.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The submodular partitioning problem asks to minimize, over all partitions P of a ground set V, the sum of a given submodular function f over the parts of P. The problem has seen considerable work in approximability, as it encompasses multiterminal cuts on graphs, k-cuts on hypergraphs, and elementary linear algebra problems such as matrix multiway partitioning. This research has been divided between the fixed terminal setting, where we are given a set of terminals that must be separated by P, and the global setting, where the only constraint is the size of the partition. We investigate a generalization that unifies these two settings: minimum submodular matroid-constrained partition. In this problem, we are additionally given a matroid over the ground set and seek to find a partition P in which there exists some basis that is separated by P. We explore the approximability of this problem and its variants, reaching the state of the art for the special case of symmetric submodular functions, and provide results for monotone and general submodular functions as well.

Problem

• Generalizes submodular partitioning with matroid constraints
• Unifies fixed terminal and global partition settings
• Advances approximability for symmetric submodular functions

Innovation

• Generalizes submodular matroid-constrained partitioning problem
• Unifies fixed terminal and global settings
• Advances approximability for symmetric submodular functions