Deterministic & Adaptive Non-Submodular Maximization via the Primal Curvature

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.

💡 Research Summary

The paper addresses a fundamental limitation of greedy algorithms: their classic approximation guarantee of (1 − 1/e) holds only for monotone submodular objective functions. Many real‑world problems are non‑submodular, and existing curvature‑based bounds (e.g., elemental curvature α) quickly degenerate to zero for such functions, offering little practical insight. To overcome this, the authors introduce a new notion called primal curvature, defined locally as

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