Locally Adaptive Optimization: Adaptive Seeding for Monotone Submodular Functions

TL;DR

This paper introduces a novel locally-adaptive policy framework that achieves a near-optimal approximation for adaptive seeding in monotone submodular functions, with implications for influence maximization and stochastic optimization.

Contribution

It presents a $(1-1/e)^2$-approximation algorithm for adaptive seeding using locally-adaptive policies, combining global non-adaptive structure with local adaptivity, and introduces a new submodular optimization problem of independent interest.

Findings

Achieves a $(1-1/e)^2$-approximation for adaptive seeding.

Introduces locally-adaptive policies that outperform non-adaptive approaches.

Identifies a fundamental submodular optimization problem with near-optimal solutions in certain cases.

Abstract

The Adaptive Seeding problem is an algorithmic challenge motivated by influence maximization in social networks: One seeks to select among certain accessible nodes in a network, and then select, adaptively, among neighbors of those nodes as they become accessible in order to maximize a global objective function. More generally, adaptive seeding is a stochastic optimization framework where the choices in the first stage affect the realizations in the second stage, over which we aim to optimize. Our main result is a $(1-1/e)^2$-approximation for the adaptive seeding problem for any monotone submodular function. While adaptive policies are often approximated via non-adaptive policies, our algorithm is based on a novel method we call \emph{locally-adaptive} policies. These policies combine a non-adaptive global structure, with local adaptive optimizations. This method enables the $(1-1/e)^2$-approximation for general monotone submodular functions and circumvents some of the impossibilities associated with non-adaptive policies. We also introduce a fundamental problem in submodular optimization that may be of independent interest: given a ground set of elements where every element appears with some small probability, find a set of expected size at most $k$ that has the highest expected value over the realization of the elements. We show a surprising result: there are classes of monotone submodular functions (including coverage) that can be approximated almost optimally as the probability vanishes. For general monotone submodular functions we show via a reduction from \textsc{Planted-Clique} that approximations for this problem are not likely to be obtainable. This optimization problem is an important tool for adaptive seeding via non-adaptive policies, and its hardness motivates the introduction of \emph{locally-adaptive} policies we use in the main result.