A LOCAL View of the Polynomial Hierarchy

Abstract: We extend classical methods of computational complexity to the setting of distributed computing, where they are sometimes more effective than in their original context. Our focus is on distributed decision in the LOCAL model, where multiple networked computers communicate via synchronous message-passing to collectively answer a question about their network topology. Rather unusually, we impose two orthogonal constraints on the running time of this model: the number of communication rounds is bounded by a constant, and the number of computation steps of each computer is polynomially bounded by the size of its local input and the messages it receives. By letting two players take turns assigning certificates to all computers in the network, we obtain a generalization of the polynomial hierarchy (and hence of the complexity classes $\mathbf{P}$ and $\mathbf{NP}$). We then extend some key results of complexity theory to this setting, in particular the Cook-Levin theorem (which identifies Boolean satisfiability as a complete problem for $\mathbf{NP}$), and Fagin's theorem (which characterizes $\mathbf{NP}$ as the problems expressible in existential second-order logic). The original results can be recovered as the special case where the network consists of a single computer. But perhaps more surprisingly, the task of separating complexity classes becomes easier in the general case: we can show that our hierarchy is infinite, while it remains notoriously open whether the same is true in the case of a single computer. (By contrast, a collapse of our hierarchy would have implied a collapse of the polynomial hierarchy.) As an application, we propose quantifier alternation as a new approach to measuring the locality of problems in distributed computing.

• The paper extends the polynomial hierarchy into distributed computing by defining a 'local-polynomial hierarchy' with certificate-based node verification.
• It adapts classical results like Cook-Levin and Fagin's theorems to the LOCAL model, proving unique separations between distributed complexity classes.
• Logical characterizations via local second-order logic validate the framework and open new avenues for efficient distributed decision-making.

An Analysis of "A LOCAL View of the Polynomial Hierarchy"

This paper presents a sophisticated exploration of the polynomial hierarchy in the context of distributed computing, extending traditional notions of computational complexity into the distributed domain. The work bridges concepts from centralized complexity theory with the specifics of distributed decision problems, particularly within Linial's LOCAL model of networked computers communicating synchronously.

Distributed Decision in the LOCAL Model

The paper focuses on distributed decision-making in the LOCAL model, where network nodes decide on properties of their topology via synchronous message-passing. Traditionally, complexity classes such as $\PTIME$ and $\NP$ are defined in a centralized context. This work adapts these concepts to distributed settings by imposing two main constraints: a constant bound on communication rounds and a polynomial bound on local computational steps relative to node inputs and received messages.

Extending the Polynomial Hierarchy

A novel contribution of the paper is extending the polynomial hierarchy (PH) to distributed settings, called the "local-polynomial hierarchy." This adaptation involves two players strategically assigning certificates to network nodes to help decide on network properties. The paper explores key aspects of complexity, such as Cook-Levin's theorem and Fagin's theorem, under this new distributed framework, extending classical results to distributed computing and demonstrating unique results due to its setting.

Local Certificate Boundedness

An essential concept introduced is the polynomially-bounded certificate assignment. These certificates help network nodes verify distributed properties within the network. The idea utilizes the locality constraint by only allowing certificates bounded by polynomial conditions relative to the neighborhood size, truly integrating both computation and locality aspects of distributed environments.

Hierarchy Separation and Locality Measures

Perhaps one of the most significant implications of this framework is the ease of separating complexity classes in a distributed setting compared to centralized models. The paper shows that the local-polynomial hierarchy is infinite, contrasting with the open question of the classical polynomial hierarchy's infiniteness. This result suggests that alternation (the back-and-forth certificate assignment) provides notable leverage in distributed computations, potentially offering a new method for measuring locality within distributed computing.

Logical Characterization and Future Prospects

The paper also provides a logical characterization of these distributed complexity classes using descriptive complexity theory. By extending Fagin's theorem to the local context, the authors show that problems verifiable by a distributed machine can be expressed in fragments of local second-order logic. This result further supports the robustness of the local-polynomial hierarchy, suggesting potential for utilizing existing logical and automata theory results to further explore and possibly solve open problems in distributed computing.

Conclusion

This work advances our understanding of computational complexity within distributed systems, providing both a theoretical foundation and practical implications for distributed algorithm design. By demonstrating the infinite nature of the local-polynomial hierarchy and leveraging logic for problem characterization, the study offers a unique perspective on the power and limitations of distributed computations, paving the way for further research into efficient distributed decision-making paradigms. It further suggests possible future directions for engaging classical complexity problems with distributed models, enhancing our overall comprehension of computation across different environments.