Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints

TL;DR

This paper introduces and analyzes two new submodular optimization problems with submodular cover and knapsack constraints, providing approximation algorithms, hardness results, and empirical validation for applications like sensor placement and data selection.

Contribution

It formulates new submodular optimization problems with submodular constraints, offers approximation algorithms with tight bounds, and demonstrates their effectiveness through experiments.

Findings

Approximation algorithms with bounded guarantees for both problems.

Hardness results showing tight bounds up to log-factors.

Empirical evidence of scalability and performance.

Abstract

We investigate two new optimization problems -- minimizing a submodular function subject to a submodular lower bound constraint (submodular cover) and maximizing a submodular function subject to a submodular upper bound constraint (submodular knapsack). We are motivated by a number of real-world applications in machine learning including sensor placement and data subset selection, which require maximizing a certain submodular function (like coverage or diversity) while simultaneously minimizing another (like cooperative cost). These problems are often posed as minimizing the difference between submodular functions [14, 35] which is in the worst case inapproximable. We show, however, that by phrasing these problems as constrained optimization, which is more natural for many applications, we achieve a number of bounded approximation guarantees. We also show that both these problems are closely related and an approximation algorithm solving one can be used to obtain an approximation guarantee for the other. We provide hardness results for both problems thus showing that our approximation factors are tight up to log-factors. Finally, we empirically demonstrate the performance and good scalability properties of our algorithms.