A New Approximation Guarantee for Monotone Submodular Function Maximization via Discrete Convexity

Abstract

In monotone submodular function maximization, approximation guarantees based on the curvature of the objective function have been extensively studied in the literature. However, the notion of curvature is often pessimistic, and we rarely obtain improved approximation guarantees, even for very simple objective functions.In this paper, we provide a novel approximation guarantee by extracting an M ♮ -concave function h : 2 E → R + , a notion in discrete convex analysis, from the objective function f : 2 E → R + . We introduce the notion of h-curvature, which measures how much f deviates from h, and show that we can obtain a (1 − γ/e − ǫ)-approximation to the problem of maximizing f under a cardinality constraint in polynomial time for any constant ǫ > 0. Then, we show that we can obtain nontrivial approximation guarantees for various problems by applying the proposed algorithm.

Submodular Maximization Under A Matroid Constraint: Asking more from an old friend, the Greedy Algorithm

“…For the special case of subset selection problem, the guarantee can be improved from (1 − 1/e) to (1 − e −c )/c (→ 1 as c → 0) (Conforti and Cornuéjols (1984)). In more recent work, (Soma and Yoshida (2017)) has improved the guarantee to (1 − γ h /e − ), where γ h is the h-curvature and for any > 0. Most of the theoretical work on the greedy algorithm assumes the value oracle model, where a polynomial algorithm is assumed to exist that can compute the optimal increment in each iteration.…”