Submodular Maximization Beyond Non-negativity: Guarantees, Fast Algorithms, and Applications

TL;DR

This paper extends submodular maximization to functions expressed as the difference of a monotone, non-negative, weakly submodular function and a non-negative modular function, providing new algorithms with strong approximation guarantees and practical applications.

Contribution

It introduces algorithms for maximizing difference-of-weakly submodular functions with provable guarantees, independent of the cardinality constraint, and demonstrates their effectiveness empirically.

Findings

Achieves approximation guarantees close to the optimal value.

Runs in time independent of the constraint size.

Successfully applied to experimental design and vertex cover problems.

Abstract

It is generally believed that submodular functions -- and the more general class of $\gamma$-weakly submodular functions -- may only be optimized under the non-negativity assumption $f(S) \geq 0$. In this paper, we show that once the function is expressed as the difference $f = g - c$, where $g$ is monotone, non-negative, and $\gamma$-weakly submodular and $c$ is non-negative modular, then strong approximation guarantees may be obtained. We present an algorithm for maximizing $g - c$ under a $k$-cardinality constraint which produces a random feasible set $S$ such that $\mathbb{E} \left[ g(S) - c(S) \right] \geq (1 - e^{-\gamma} - \epsilon) g(OPT) - c(OPT)$, whose running time is $O (\frac{n}{\epsilon} \log^2 \frac{1}{\epsilon})$, i.e., independent of $k$. We extend these results to the unconstrained setting by describing an algorithm with the same approximation guarantees and faster $O(\frac{n}{\epsilon} \log\frac{1}{\epsilon})$ runtime. The main techniques underlying our algorithms are two-fold: the use of a surrogate objective which varies the relative importance between $g$ and $c$ throughout the algorithm, and a geometric sweep over possible $\gamma$ values. Our algorithmic guarantees are complemented by a hardness result showing that no polynomial-time algorithm which accesses $g$ through a value oracle can do better. We empirically demonstrate the success of our algorithms by applying them to experimental design on the Boston Housing dataset and directed vertex cover on the Email EU dataset.