The Cost of Consistency: Submodular Maximization with Constant Recourse

TL;DR

This paper investigates the limits of online submodular maximization under stability constraints, establishing tight bounds for approximation ratios and providing a practical randomized algorithm with guaranteed performance.

Contribution

It derives tight information-theoretic bounds for monotone submodular maximization with limited updates and introduces a polynomial-time randomized algorithm achieving a 0.51 approximation.

Findings

Tight bound of 2/3 for general monotone submodular functions.

Improved tight bound of 3/4 for coverage functions.

A polynomial-time randomized algorithm with 0.51 approximation ratio.

Abstract

In this work, we study online submodular maximization, and how the requirement of maintaining a stable solution impacts the approximation. In particular, we seek bounds on the best-possible approximation ratio that is attainable when the algorithm is allowed to make at most a constant number of updates per step. We show a tight information-theoretic bound of $\tfrac{2}{3}$ for general monotone submodular functions, and an improved (also tight) bound of $\tfrac{3}{4}$ for coverage functions. Since both these bounds are attained by non poly-time algorithms, we also give a poly-time randomized algorithm that achieves a $0.51$-approximation. Combined with an information-theoretic hardness of $\tfrac{1}{2}$ for deterministic algorithms from prior work, our work thus shows a separation between deterministic and randomized algorithms, both information theoretically and for poly-time algorithms.