Stochastic Multi-round Submodular Optimization with Budget

TL;DR

This paper addresses the complex problem of multi-round stochastic submodular maximization under budget constraints, proposing a polynomial-time dynamic programming solution, a greedy approximation algorithm, and analyzing the benefits of adaptivity.

Contribution

It introduces a novel greedy approximation algorithm for SBMSm, extends the problem to multiple rounds, and defines the budget-adaptivity gap to quantify adaptive policy advantages.

Findings

Polynomial-time dynamic programming algorithm for bounded instances.

Greedy 1/2(1-1/e-ε) approximation algorithm.

Budget-adaptivity gap between 1.582 and 2.

Abstract

In this work, we study the Stochastic Budgeted Multi-round Submodular Maximization (SBMSm) problem, where we aim to adaptively maximize the sum, over multiple rounds, of a monotone and submodular objective function defined on subsets of items. The objective function also depends on the realization of stochastic events, and the total number of items we can select over all rounds is bounded by a limited budget. This problem extends, and generalizes to multiple round settings, well-studied problems such as (adaptive) influence maximization and stochastic probing. We show that, if the number of items and stochastic events is somehow bounded, there is a polynomial time dynamic programming algorithm for SBMSm. Then, we provide a simple greedy $1/2(1-1/e-\epsilon)\approx 0.316$-approximation algorithm for SBMSm, that first non-adaptively allocates the budget to be spent at each round, and then greedily and adaptively maximizes the objective function by using the budget assigned at each round. Finally, we introduce the {\em budget-adaptivity gap}, by which we measure how much an adaptive policy for SBMSm is better than an optimal partially adaptive one that, as in our greedy algorithm, determines the budget allocation in advance. We show that the budget-adaptivity gap lies between $e/(e-1)\approx 1.582$ and $2$.