Optimally Controlling a Random Population

TL;DR

This paper proves that the problem of controlling a large population of identical agents modeled as Markov Decision Processes, to reach a target state, is EXPTIME-complete, establishing its computational complexity.

Contribution

It establishes the exact computational complexity (EXPTIME-complete) of the randomized population control problem, improving understanding of its decidability and complexity bounds.

Findings

The problem is decidable.

It is EXPTIME-complete.

Provides complexity bounds for population control scenarios.

Abstract

The population control problem is a parameterised problem where a controller sends messages to a whole population of identical finite-state agents, aiming to eventually move them all into a target state. The decision problem asks whether this can be achieved for arbitrarily large finite populations. We focus on the randomised version of this problem, where every agent is a copy of the same finite Markov Decision Process and non-determinism in the {global} action chosen by the controller is resolved independently and uniformly at random. Colcombet, Fijalkow and Ohlmann showed that this problem is decidable, but without any complexity upper bound. We show that the random population control problem is in fact EXPTIME-complete.