Graph Exploration: The Impact of a Distance Constraint

TL;DR

This paper investigates the limitations of deterministic exploration algorithms in graphs under distance constraints, proving that linear penalty guarantees are impossible, thus highlighting fundamental differences from unconstrained exploration.

Contribution

It proves that no universal exploration algorithm can guarantee a linear penalty under distance constraints, solving an open problem and revealing a fundamental separation from unconstrained exploration.

Findings

Linear penalty algorithms are impossible under distance constraints.

Distance-constrained exploration differs fundamentally from unconstrained exploration.

The results answer an open problem posed by Duncan et al.

Abstract

A mobile agent, starting from a node $s$ of a simple undirected connected graph $G=(V,E)$, has to explore all nodes and edges of $G$ using the minimum number of edge traversals. To do so, the agent uses a deterministic algorithm that allows it to gain information on $G$ as it traverses its edges. During its exploration, the agent must always respect the constraint of knowing a path of length at most $D$ to go back to node $s$. The upper bound $D$ is fixed as being equal to $(1+\alpha)r$, where $r$ is the eccentricity of node $s$ (i.e., the maximum distance from $s$ to any other node) and $\alpha$ is any positive real constant. This task has been introduced by Duncan et al. [ACM Trans. Algorithms 2006] and is known as \emph{distance-constrained exploration}. The \emph{penalty} of an exploration algorithm running in $G$ is the number of edge traversals made by the agent in excess of $|E|$. Panaite and Pelc [J. Algorithms 1999] gave an algorithm for solving exploration without any constraint on the moves that is guaranteed to work in every graph $G$ with a (small) penalty in $\mathcal{O}(|V|)$. Hence, a natural question is whether we could obtain a distance-constrained exploration algorithm with the same guarantee as well. In this paper, we provide a negative answer to this question. We also observe that an algorithm working in every graph $G$ with a linear penalty in $|V|$ cannot be obtained for the task of \emph{fuel-constrained exploration}, another variant studied in the literature. This solves an open problem posed by Duncan et al. [ACM Trans. Algorithms 2006] and shows a fundamental separation with the task of exploration without constraint on the moves.