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arXiv · 1702.07002

Deterministic & Adaptive Non-Submodular Maximization via the Primal Curvature

J. David Smith·My T. Thai
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Abstract

While greedy algorithms have long been observed to perform well on a wide variety of problems, up to now approximation ratios have only been known for their application to problems having submodular objective functions $f$. Since many practical problems have non-submodular $f$, there is a critical need to devise new techniques to bound the performance of greedy algorithms in the case of non-submodularity. Our primary contribution is the introduction of a novel technique for estimating the approximation ratio of the greedy algorithm for maximization of monotone non-decreasing functions based on the curvature of $f$ without relying on the submodularity constraint. We show that this technique reduces to the classical $(1 - 1/e)$ ratio for submodular functions. Furthermore, we develop an extension of this ratio to the adaptive greedy algorithm, which allows applications to non-submodular stochastic maximization problems. This notably extends support to applications modeling incomplete data with uncertainty.

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J. David Smith, My T. Thai. 2017-02-22. Deterministic & Adaptive Non-Submodular Maximization via the Primal Curvature. https://arxiv.org/abs/1702.07002

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