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 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 approxi- mated almost optimally as the probability vanishes. For general monotone submodular functions we show via a reduction from 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 locally-adaptive policies we use in the main result. We use the following bound in the paper (See for example [30, Theorem 4.4]). Theorem A.1. (Chernoff bound) Suppose X 1 , . . . , X n are independent random variables taking values in [0, 1] and let X denote their sum and µ be the expected value of X. Then, for 0 < δ < 1 we have that P r[X > (1 + δ)µ] ≤ exp(− δ 2 µ 3 ) 429
Badanidiyuru et al. REF propose an algorithm based on locally-adaptive policies.
Locally Adaptive Optimization: Adaptive Seeding for Monotone Submodular Functions
Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms
Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms
Computer Science