The Recursive Arrival Problem

TL;DR

This paper introduces Recursive Arrival, an extension of the Arrival problem involving recursive switching graphs, and analyzes its computational complexity, showing it lies in NP ∩ coNP and is P-hard, with connections to complexity classes like UEOPL and EOPL.

Contribution

It extends the Arrival problem to recursive graphs and provides complexity bounds, including containment in NP ∩ coNP and P-hardness, with implications for related complexity classes.

Findings

Recursive Arrival is in NP ∩ coNP.

A search version of Recursive Arrival is in UEOPL, and thus in EOPL.

Recursive Arrival decision problem is P-hard.

Abstract

We study an extension of the Arrival problem, called Recursive Arrival, inspired by Recursive State Machines, which allows for a family of switching graphs that can call each other in a recursive way. We study the computational complexity of deciding whether a Recursive Arrival instance terminates at a given target vertex. We show this problem is contained in NP \cap coNP, and we show that a search version of the problem lies in UEOPL, and hence in EOPL = PLS \cap PPAD. Furthermore, we show P-hardness of the Recursive Arrival decision problem. By contrast, the current best-known hardness result for Arrival is PL-hardness.