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Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints (2013)
| Citations: | 14 - 8 self |
BibTeX
@MISC{Iyer13submodularoptimization,
author = {Rishabh Iyer and Jeff Bilmes},
title = {Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints },
year = {2013}
}
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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 [9, 25] 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.
Keyphrases
submodular cover arxiv version submodular optimization submodular knapsack constraint submodular function subject many application hardness result constrained optimization cooperative cost new optimization problem bounded approximation guarantee good scalability property bound constraint data subset selection submodular knapsack certain submodular function approximation algorithm submodular upper bound constraint submodular function approximation guarantee approximation factor real-world application sensor placement
