Generalizing Performance Bounds for the Greedy Algorithm: Approximate Submodularity and Localization
Year: 2019
Type: preprint
Abstract: Submodularity, a discrete analog of convexity, is a key property in discrete optimization that features in the construction of valid inequalities and analysis of the greedy algorithm. In this paper, we broaden the approximate submodularity literature, which so far has largely focused on variants of greedy algorithms and iterative approaches. We define metrics that quantify approximate submodularity and use these metrics to derive properties about approximate submodularity preservation and extensions of set functions. We show that previous analyses of mixed-integer sets, such as the submodular knapsack polytope, can be extended to the approximate submodularity setting. In addition, we demonstrate that greedy algorithm bounds based on our notions of approximate submodularity are competitive with those in the literature, which we illustrate using a generalization of the uncapacitated facility location problem.
Source: arXiv (Cornell University)
Authors: Temitayo Ajayi, Taewoo Lee, Andrew J. Schaefer
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2913973277