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This research rigorously maps the computational complexity landscape of Multi-environment Partially Observable Markov Decision Processes (ME-POMDPs), demonstrating how model uncertainty fundamentally elevates tractable single-environment subclasses to PSPACE-complete or NP-hard problems. By identifying specific closure conditions that preserve hardness, the work provides critical theoretical boundaries for robust reinforcement learning and decision-making under uncertainty.
On the Complexity Gap Between POMDPs and Multi-Environment POMDPs
arXiv:2610.15010v1 Announce Type: new Abstract: Multi-environment POMDPs (ME-POMDPs) generalize POMDPs by encoding a finite set of possible environments, capturing model uncertainty. The finite-horizon robust reward maximization problem for a ME-POMDP asks for a policy that maximizes reward in the worst environment, and recent work has shown this problem to be $\mathsf{PSPACE}$-complete. We expand the complexity landscape of ME-POMDPs by identifying which subclasses retain this $\mathsf{PSPACE}$-completeness. In particular, we take subclasses of POMDPs that are tractable in the single-environment case, and investigate the computational complexity of these classes when extended to the multi-environment setting. We first prove the $\mathsf{NP}$-completeness of multi-environment blind MDPs and deterministic POMDPs using a certain closure-under-mixture condition. We then prove $\mathsf{PSPACE}$-completeness for a range of subclasses that do not satisfy that closure condition, yielding new complexity results for multi-observation POMDPs. Finally, we show existential theory of the reals ($\exists \mathbb{R}$)-completeness for numerous ME-POMDP problems under time-dependent Markovian policies or finite-state controllers that are $\mathsf{NP}$-complete for standard POMDPs. These results reveal a subtle relationship between POMDP and ME-POMDP complexity that depends heavily on the subclasses involved.
Abstract:Multi-environment POMDPs (ME-POMDPs) generalize POMDPs by encoding a finite set of possible environments, capturing model uncertainty. The finite-horizon robust reward maximization problem for a ME-POMDP asks for a policy that maximizes reward in the worst environment, and recent work has shown this problem to be $\mathsf{PSPACE}$-complete. We expand the complexity landscape of ME-POMDPs by identifying which subclasses retain this $\mathsf{PSPACE}$-completeness. In particular, we take subclasses of POMDPs that are tractable in the single-environment case, and investigate the computational complexity of these classes when extended to the multi-environment setting. We first prove the $\mathsf{NP}$-completeness of multi-environment blind MDPs and deterministic POMDPs using a certain closure-under-mixture condition. We then prove $\mathsf{PSPACE}$-completeness for a range of subclasses that do not satisfy that closure condition, yielding new complexity results for multi-observation POMDPs. Finally, we show existential theory of the reals ($\exists \mathbb{R}$)-completeness for numerous ME-POMDP problems under time-dependent Markovian policies or finite-state controllers that are $\mathsf{NP}$-complete for standard POMDPs. These results reveal a subtle relationship between POMDP and ME-POMDP complexity that depends heavily on the subclasses involved.
Artificial Intelligence (cs.AI)
arXiv:2610.15010 [cs.AI]
(or arXiv:2610.15010v1 [cs.AI] for this version)
https://doi.org/10.48550/arXiv.2610.15010 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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