ARRIVAL: Recursive Framework&$ell_1$-Contraction

2025-02-10

AI Summary

The ARRIVAL problem asks whether a token traversing a directed graph according to switching rules reaches one of two designated targets first; its complexity lies in NP ∩ coNP, and membership in P remains open. This paper focuses on graphs with treewidth k and n vertices, presenting the first treewidth-parameterized subexponential algorithm, running in time 2^{O(k log² n)}. Methodologically, we establish an equivalence between ARRIVAL and approximating fixed points of ℓ₁-contractive functions over [0,1]ⁿ, uncovering a deep structural analogy with simple stochastic games (SSGs). Building on this insight, we introduce a recursive framework and a generalized model—G-ARRIVAL—which unifies and reproves Gärtner et al.’s classic √n·log n upper bound. Our approach shifts the analytical paradigm from purely graph-theoretic reasoning to metric-space analysis, enabling tighter parameterized bounds and revealing new connections between combinatorial optimization and fixed-point computation.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

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 $2^{mathcal{O}(sqrt{n} log n)}$ on an $n$-vertex graph. We prove that ARRIVAL can be solved in time $2^{mathcal{O}(k log^2 n)}$ 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 $f : [0, 1]^n ightarrow [0, 1]^n$. 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.

Problem

• Determining token destination in directed graphs
• Solving ARRIVAL problem with treewidth parameter
• Reducing G-ARRIVAL to ℓ1-contracting fixed points

Innovation

• Recursive algorithm framework
• Treewidth parameterization technique
• ℓ₁-contraction fixed point reduction