The Complexity of Decentralized Control of Markov Decision Processes (1301.3836v1)

Abstract: Planning for distributed agents with partial state information is considered from a decision- theoretic perspective. We describe generalizations of both the MDP and POMDP models that allow for decentralized control. For even a small number of agents, the finite-horizon problems corresponding to both of our models are complete for nondeterministic exponential time. These complexity results illustrate a fundamental difference between centralized and decentralized control of Markov processes. In contrast to the MDP and POMDP problems, the problems we consider provably do not admit polynomial-time algorithms and most likely require doubly exponential time to solve in the worst case. We have thus provided mathematical evidence corresponding to the intuition that decentralized planning problems cannot easily be reduced to centralized problems and solved exactly using established techniques.

• The paper proves that finite-horizon DEC-POMDPs (for m ≥ 2) and DEC-MDPs (for m ≥ 3) are NEXP-complete, highlighting extreme computational complexity.
• The authors employ a reduction from the NEXP-complete TILING problem and a nondeterministic guess-and-check method to rigorously establish complexity bounds.
• These findings imply that decentralized multi-agent systems require novel algorithmic approaches, as traditional centralized methods are insufficient.

The Complexity of Decentralized Control of Markov Decision Processes

The paper "The Complexity of Decentralized Control of Markov Decision Processes" by Daniel S. Bernstein, Shlomo Zilberstein, and Neil Immerman provides a detailed examination of decentralized control mechanisms within the context of Markov Decision Processes (MDPs) and their partially observable counterparts (POMDPs). It contributes significantly to the computational complexity theory related to distributed agents operating under partial state information.

Overview

Centralized planning approaches for MDPs are well-characterized and known to be P-complete for deterministic polynomial-time constraint satisfaction, while the corresponding issues in POMDPs extend to PSPACE-completeness, reflecting a higher computational demand due to incomplete information. The authors extend this analysis to scenarios necessitating decentralized control, introducing the decentralized partially observable Markov decision process (DEC-POMDP) and the decentralized Markov decision process (DEC-MDP).

Key Findings

For finite-horizon problems, the authors establish that both DEC-POMDPs and DEC-MDPs exhibit considerably higher computational complexity compared to their centralized counterparts:

• Computational Complexity: The problems are proven to be complete for nondeterministic exponential time (NEXP). Specifically, solving a DEC-POMDP with a constant number of agents $m \geq 2$ is NEXP-complete. Similarly, solving a DEC-MDP for $m \geq 3$ achieves the same complexity classification.
• Worst-case Time Complexity: These results imply that DEC-POMDPs and DEC-MDPs do not admit polynomial-time algorithms. Additionally, the most efficient known algorithms for these problems likely require doubly exponential time in the worst case, thereby starkly distinguishing decentralized problems from their centralized analogues.
• Implications on Reductions: The findings provide rigorous evidence supporting the intuition that decentralized planning problems cannot be simply reduced to centralized problems. Consequently, traditional centralized methods and reductions are not applicable, steering research towards fundamentally different algorithmic paradigms for decentralized control.

Methodology

The authors leverage the TILING problem, which is known to be NEXP-complete, as a foundation for their complexity proofs. By constructing an isomorphic problem within the DEC-POMDP framework, they effectively demonstrate the equivalency in complexity. The DEC-POMDP problem formulation involves stipulating that each agent only has a local view of the state through sensory input, and decisions are made based on these observations which collectively determine state transitions and rewards.

The proof for inclusion in NEXP follows standard guess-and-check approaches in nondeterministic computational settings, while the proof for NEXP-hardness utilizes an intricate reduction from the TILING problem, preserving key characteristics such as state transition dependencies and observation hierarchies.

Theoretical and Practical Implications

These complexity results have profound implications:

• Algorithm Development: The exponential increase in complexity highlights that existing methods for POMDPs cannot be directly adapted or extended to solve DEC-POMDPs or DEC-MDPs. New algorithms that can handle the decentralized nature of these problems must be developed, potentially incorporating approximation strategies or heuristic methods.
• Distributed Systems: In practical applications, such as multi-robot coordination or distributed network control, these findings underscore the inherent difficulties posed by decentralized information and control. Efficient and scalable solutions in these domains must account for the significant computational overhead.
• Complexity Theory: The paper draws connections to broader topics in complexity theory, emphasizing the significant computational leaps introduced by decentralized decision-making processes. It also opens up further questions regarding specific bounds and classifications for various agent numbers and observation models.

Future Directions

Future research could explore several avenues:

• Approximation Algorithms: Given the infeasibility of exact solutions in reasonable time frames, developing robust approximation techniques could provide practical benefits.
• Policy Space Exploration: Techniques focusing on directly searching through policy spaces rather than state spaces may offer more scalable solutions.
• Comparison to Infinite-horizon Problems: Extending the complexity analysis to infinite-horizon versions and comparing decidability results could provide deeper insights into long-term planning for decentralized systems.

Overall, Bernstein, Zilberstein, and Immerman's paper makes a substantial contribution to the understanding of decentralized control in MDP frameworks. By rigorously analyzing the computational complexity of DEC-POMDPs and DEC-MDPs, the authors provide a critical foundation for future research aimed at tackling distributed decision-making challenges in AI and beyond.