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Performance Bounds with Curvature for Batched Greedy Optimization

  • Yajing LiuEmail author
  • Zhenliang Zhang
  • Edwin K. P. Chong
  • Ali Pezeshki
Article

Abstract

The batched greedy strategy is an approximation algorithm to maximize a set function subject to a matroid constraint. Starting with the empty set, the batched greedy strategy iteratively adds to the current solution set a batch of elements that results in the largest gain in the objective function while satisfying the matroid constraints. In this paper, we develop bounds on the performance of the batched greedy strategy relative to the optimal strategy in terms of a parameter called the total batched curvature. We show that when the objective function is a polymatroid set function, the batched greedy strategy satisfies a harmonic bound for a general matroid constraint and an exponential bound for a uniform matroid constraint, both in terms of the total batched curvature. We also study the behavior of the bounds as functions of the batch size. Specifically, we prove that the harmonic bound for a general matroid is nondecreasing in the batch size and the exponential bound for a uniform matroid is nondecreasing in the batch size under the condition that the batch size divides the rank of the uniform matroid. Finally, we illustrate our results by considering a task scheduling problem and an adaptive sensing problem.

Keywords

Curvature Greedy Matroid Polymatroid Submodular 

Mathematics Subject Classification

90C27 90C59 

Notes

Acknowledgements

This work is supported in part by NSF under award CCF-1422658 and by the CSU Information Science and Technology Center (ISTeC). We would like to acknowledge the anonymous reviewers for their insightful comments on our conference paper [26], as these comments led us to improve our work.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Colorado State UniversityFort CollinsUSA
  2. 2.Intel LabsHillsboroUSA

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