ARRIVAL: Next Stop in CLS

TL;DR

This paper advances the understanding of the ARRIVAL game by placing its decision and search variants into complexity classes UP∩coUP and CLS respectively, and provides a faster randomized algorithm with sub-exponential runtime.

Contribution

It improves upper bounds for ARRIVAL's decision and search problems, showing they are in UP∩coUP and CLS, and introduces a sub-exponential randomized algorithm.

Findings

ARRIVAL decision problem is in UP∩coUP.

Search variant of ARRIVAL is in CLS.

Introduces a randomized algorithm with expected runtime O(1.4143^n).

Abstract

We study the computational complexity of ARRIVAL, a zero-player game on $n$-vertex switch graphs introduced by Dohrau, G\"{a}rtner, Kohler, Matou\v{s}ek, and Welzl. They showed that the problem of deciding termination of this game is contained in $\text{NP} \cap \text{coNP}$. Karthik C. S. recently introduced a search variant of ARRIVAL and showed that it is in the complexity class PLS. In this work, we significantly improve the known upper bounds for both the decision and the search variants of ARRIVAL. First, we resolve a question suggested by Dohrau et al. and show that the decision variant of ARRIVAL is in $\text{UP} \cap \text{coUP}$. Second, we prove that the search variant of ARRIVAL is contained in CLS. Third, we give a randomized $\mathcal{O}(1.4143^n)$-time algorithm to solve both variants. Our main technical contributions are (a) an efficiently verifiable characterization of the unique witness for termination of the ARRIVAL game, and (b) an efficient way of sampling from the state space of the game. We show that the problem of finding the unique witness is contained in CLS, whereas it was previously conjectured to be FPSPACE-complete. The efficient sampling procedure yields the first algorithm for the problem that has expected runtime $\mathcal{O}(c^n)$ with $c<2$.