Guarantees for Greedy Maximization of Non-submodular Functions with Applications

TL;DR

This paper provides theoretical guarantees for the performance of the Greedy algorithm when maximizing non-submodular, nondecreasing set functions under cardinality constraints, supported by empirical validation.

Contribution

It introduces a new approximation guarantee based on curvature and submodularity ratio, extending theoretical understanding to non-submodular functions.

Findings

Greedy achieves a tight approximation ratio of 1/α(1 - e^{-γα})

Bounds on submodularity ratio and curvature for key real-world objectives

Empirical validation confirms theoretical guarantees

Abstract

We investigate the performance of the standard Greedy algorithm for cardinality constrained maximization of non-submodular nondecreasing set functions. While there are strong theoretical guarantees on the performance of Greedy for maximizing submodular functions, there are few guarantees for non-submodular ones. However, Greedy enjoys strong empirical performance for many important non-submodular functions, e.g., the Bayesian A-optimality objective in experimental design. We prove theoretical guarantees supporting the empirical performance. Our guarantees are characterized by a combination of the (generalized) curvature $\alpha$ and the submodularity ratio $\gamma$. In particular, we prove that Greedy enjoys a tight approximation guarantee of $\frac{1}{\alpha}(1- e^{-\gamma\alpha})$ for cardinality constrained maximization. In addition, we bound the submodularity ratio and curvature for several important real-world objectives, including the Bayesian A-optimality objective, the determinantal function of a square submatrix and certain linear programs with combinatorial constraints. We experimentally validate our theoretical findings for both synthetic and real-world applications.