On Polynomial Time Local Decision

The field of distributed local decision studies the power of local network algorithms, where each network can see only its own local neighborhood, and must act based on this restricted information. Traditionally, the nodes of the network are assumed to have unbounded local computation power, and this makes the model incomparable with centralized notions of efficiency, namely, the classes P and NP. In this work we seek to bridge this gap by studying local algorithms where the nodes are required to be computationally efficient: we introduce the classes PLD and NPLD of polynomial-time local decision and non-deterministic polynomial-time local decision, respectively, and compare them to the centralized complexity classes P and NP, and to the distributed classes LD and NLD, which correspond to local deterministic and non-deterministic decision, respectively. We show that for deterministic algorithms, requiring both computational and distributed efficiency is likely to be more restrictive than either requirement alone: if the nodes do not know the network size, then PLD ⊆ LD ∩ P holds unconditionally; if the network size is known to all nodes, then the same separation holds under a widely believed complexity assumption (UP ∩ coUP ≠ P). However, when nondeterminism is introduced, this distinction vanishes, and NPLD = NLD ∩ NP. To complete the picture, we extend the classes PLD and NPLD into a hierarchy akin to the centralized polynomial hierarchy, and we characterize its connections to the centralized polynomial hierarchy and to the distributed local decision hierarchy of Balliu, D’Angelo, Fraigniaud, and Olivetti.