A Multi-Arm Bandit Approach To Subset Selection Under Constraints

We explore the class of problems where a central planner needs to select a subset of agents, each with different quality and cost. The planner wants to maximize its utility while ensuring that the average quality of the selected agents is above a certain threshold. When the agents' quality is known, we formulate our problem as an integer linear program (ILP) and propose a deterministic algorithm, namely DPSS that provides an exact solution to our ILP. We then consider the setting when the qualities of the agents are unknown. We model this as a Multi-Arm Bandit (MAB) problem and propose DPSS-UCB to learn the qualities over multiple rounds. We show that after a certain number of rounds, 𝜏, DPSS-UCB outputs a subset of agents that satisfy the average quality constraint with a high probability. Next, we provide bounds on 𝜏 and prove that after 𝜏 rounds, the algorithm incurs a regret of 𝑂 (ln𝑇), where 𝑇 is the total number of rounds. We further illustrate the efficacy of DPSS-UCB through simulations. To overcome the computational limitations of DPSS, we propose a polynomial-time greedy algorithm, namely GSS, that provides an approximate solution to our ILP. We also compare the performance of DPSS and GSS through experiments.