Polynomial Time Local Decision Revisited

2026-03-31

AI Summary

This work investigates the impact of local computation and certificate models on the expressive power of distributed decision tasks. By systematically comparing three models—certificate-free, existential certificates, and universal certificates—under polynomial-time bounds defined either by global graph size or local neighborhood size, the study reveals fundamental differences in their ability to characterize decidable language classes. The main contributions include proving the equivalence of the two polynomial-time variants in the certificate-free setting, establishing a strict hierarchy under existential certificates, and uncovering counterintuitive incomparability relations under universal certificates. These results collectively yield a complete characterization of the intricate inclusion and separation structure among the three models, providing the first comprehensive lattice-theoretic depiction of the foundational complexity classes in distributed decision.

Abstract

We consider three classification systems for distributed decision tasks: With unbounded computation and certificates, defined by Balliu, D'Angelo, Fraigniaud, and Olivetti [JCSS'18], and with (two flavors of) polynomially bounded local computation and certificates, defined in recent works by Aldema Tshuva and Oshman [OPODIS'23], and by Reiter [PODC'24]. The latter two differ in the way they evaluate the polynomial bounds: the former considers polynomials with respect to the size of the graph, while the latter refers to being polynomial in the size of each node's local neighborhood. We start by revisiting decision without certificates. For this scenario, we show that the latter two definitions coincide: roughly, a node cannot know the graph size, and thus can only use a running time dependent on its neighborhood. We then consider decision with certificates. With existential certificates ($Σ_1$-type classes), a larger running time defines strictly larger classes of languages: when it grows from being polynomial in each node's view, through polynomial in the graph's size, and to unbounded, the derived classes strictly contain each other. With universal certificates ($Π_1$-type classes), on the other hand, we prove a surprising incomparability result: having running time bounded by the graph size sometimes allows us to decide languages undecidable even with unbounded certificates. We complement these results with other containment and separation results, which together portray a surprisingly complex lattice of strict containment relations between the classes at the base of the three classification systems.

Problem

• distributed decision
• local computation
• certificates
• complexity classes
• polynomial time

Innovation

• distributed decision
• local computation
• certificate complexity
• polynomial time
• Σ₁/Π₁ classes