ARRIVAL: Recursive Framework & $\ell_1$-Contraction

TL;DR

This paper introduces a recursive framework for ARRIVAL, improving its complexity on graphs with bounded treewidth, and connects ARRIVAL to fixed point problems of $ ext{l}_1$-contraction, revealing parallels with stochastic games.

Contribution

It presents a treewidth-dependent algorithm for ARRIVAL and establishes a reduction to $ ext{l}_1$-fixed point computation, extending understanding of ARRIVAL's complexity and its relation to contraction problems.

Findings

ARRIVAL can be solved in $2^{O(k ext{log}^2 n)}$ time for graphs of treewidth $k$.

A reduction from G-ARRIVAL to $ ext{l}_1$-fixed point finding is established.

The results reveal parallels between ARRIVAL and Simple Stochastic Games.

Abstract

ARRIVAL is the problem of deciding which out of two possible destinations will be reached first by a token that moves deterministically along the edges of a directed graph, according to so-called switching rules. It is known to lie in NP $\cap$ CoNP, but not known to lie in P. The state-of-the-art algorithm due to G\"artner et al. (ICALP `21) runs in time $2^{O(\sqrt{n} \log n)}$ on an $n$-vertex graph. We prove that ARRIVAL can be solved in time $2^{O(k \log^2 n)}$ on $n$-vertex graphs of treewidth $k$. Our algorithm is derived by adapting a simple recursive algorithm for a generalization of ARRIVAL called G-ARRIVAL. This simple recursive algorithm acts as a framework from which we can also rederive the subexponential upper bound of G\"artner et al. Our second result is a reduction from G-ARRIVAL to the problem of finding an approximate fixed point of an $\ell_1$-contracting function $f : [0, 1]^n \rightarrow [0, 1]^n$. Finding such fixed points is a well-studied problem in the case of the $\ell_2$-metric and the $\ell_\infty$-metric, but little is known about the $\ell_1$-case. Both of our results highlight parallels between ARRIVAL and the Simple Stochastic Games (SSG) problem. Concretely, Chatterjee et al. (SODA `23) gave an algorithm for SSG parameterized by treewidth that achieves a similar bound as we do for ARRIVAL, and SSG is known to reduce to $\ell_\infty$-contraction.