Adaptive Greedy versus Non-adaptive Greedy for Influence Maximization

Abstract

We consider the adaptive influence maximization problem: given a network and a budget k, iteratively select k seeds in the network to maximize the expected number of adopters. In the full-adoption feedback model, after selecting each seed, the seed-picker observes all the resulting adoptions. In the myopic feedback model, the seed-picker only observes whether each neighbor of the chosen seed adopts. Motivated by the extreme success of greedy-based algorithms/heuristics for influence maximization, we propose the concept of greedy adaptivity gap, which compares the performance of the adaptive greedy algorithm to its non-adaptive counterpart. Our first result shows that, for submodular influence maximization, the adaptive greedy algorithm can perform up to a (1 − 1/e)-fraction worse than the non-adaptive greedy algorithm, and that this ratio is tight. More specifically, on one side we provide examples where the performance of the adaptive greedy algorithm is only a (1−1/e) fraction of the performance of the non-adaptive greedy algorithm in four settings: for both feedback models and both the independent cascade model and the linear threshold model. On the other side, we prove that in any submodular cascade, the adaptive greedy algorithm always outputs a (1 − 1/e)-approximation to the expected number of adoptions in the optimal non-adaptive seed choice. Our second result shows that, for the general submodular diffusion model with full-adoption feedback, the adaptive greedy algorithm can outperform the non-adaptive greedy algorithm by an unbounded factor. Finally, we propose a risk-free variant of the adaptive greedy algorithm that always performs no worse than the non-adaptive greedy algorithm.

Better Bounds on the Adaptivity Gap of Influence Maximization under Full-adoption Feedback

“…Gupta et al Singla 2016, 2017) worked on the adaptivity gaps for stochastic probing. A recent line of studies has been conducted (Chen and Peng 2019;Chen et al 2020;Peng and Chen 2019) which focuses on finding the adaptivity gaps on different graph classes using the classical feedback models. Peng and Chen (2019) confirmed a conjecture of Golovin and Krause (2011), which states that the adaptive greedy algorithm with myopic feedback is a constant approximation of the adaptive optimal solution.…”

“…They show that the adaptivity gap of the independent cascade model with myopic feedback belongs to [ −1 , 4]. Chen et al (2020) introduced the greedy adaptivity gap, which compares the performance of the adaptive and the non-adaptive greedy algorithms. They show that the infimum of the greedy adaptivity gap is 1 − 1 for every combination of diffusion and feedback models.…”