Stochastic Multi-round Submodular Optimization with Budget

In this work we study the problem of Stochastic Budgeted Multi-round Submodular Maximization (SBMSm), in which we would like to maximize the sum over multiple rounds of the value of a monotone and submodular objective function, subject to the fact that the values of this function depend on the realization of stochastic events and the number of observations that we can make over all rounds is limited by a given budget. This problem extends, and generalizes to multiple round settings, well-studied problems such as (adaptive) influence maximization and stochastic probing. We first show that whenever a certain single-round optimization problem can be optimally solved in polynomial time, then there is a polynomial time dynamic programming algorithm that returns the same solution as the optimal algorithm, that can adaptively choose both which observations to make and in which round to have them. Unfortunately, this dynamic programming approach cannot be extended to work when the single-round optimization problem cannot be efficiently solved (even if we allow it would be approximated within an arbitrary small constant). Anyway, in this case we are able to provide a simple greedy algorithm for the problem. It guarantees a $(1/2-\epsilon)$-approximation to the optimal value, even if it non-adaptively allocates the budget to rounds.

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

In the influence maximization (IM) problem, we are given a social network and a budget $k$, and we look for a set of $k$ nodes in the network, called seeds, that maximize the expected number of nodes that are reached by an influence cascade generated by the seeds, according to some stochastic model for influence diffusion. In this paper, we study the adaptive IM, where the nodes are selected sequentially one by one, and the decision on the $i$th seed can be based on the observed cascade produced by the first $i-1$ seeds. We focus on the full-adoption feedback in which we can observe the entire cascade of each previously selected seed and on the independent cascade model where each edge is associated with an independent probability of diffusing influence. Our main result is the first sub-linear upper bound that holds for any graph. Specifically, we show that the adaptivity gap is upper-bounded by $\lceil n^{1/3}\rceil $, where $n$ is the number of nodes in the graph. Moreover, we improve over the known upper bound for in-arborescences from $\frac{2e}{e-1}\approx 3.16$ to $\frac{2e^2}{e^2-1}\approx 2.31$. Finally, we study $\alpha$-bounded graphs, a class of undirected graphs in which the sum of node degrees higher than two is at most $\alpha$, and show that the adaptivity gap is upper-bounded by $\sqrt{\alpha}+O(1)$. Moreover, we show that in 0-bounded graphs, i.e. undirected graphs in which each connected component is a path or a cycle, the adaptivity gap is at most $\frac{3e^3}{e^3-1}\approx 3.16$. To prove our bounds, we introduce new techniques to relate adaptive policies with non-adaptive ones that might be of their own interest.