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Graph Exploration: The Impact of a Distance Constraint

Stéphane Devismes, Yoann Dieudonné, Arnaud Labourel

TL;DR

It is observed 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.

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 $(1+α)r$, where $r$ is the eccentricity of node $s$ (i.e., the maximum distance from $s$ to any other node) and $α$ 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.

Graph Exploration: The Impact of a Distance Constraint

TL;DR

It is observed that an algorithm working in every graph with a linear penalty in cannot be obtained for the task of \emph{fuel-constrained exploration}, another variant studied in the literature.

Abstract

A mobile agent, starting from a node of a simple undirected connected graph , has to explore all nodes and edges of using the minimum number of edge traversals. To do so, the agent uses a deterministic algorithm that allows it to gain information on as it traverses its edges. During its exploration, the agent must always respect the constraint of knowing a path of length at most to go back to node . The upper bound is fixed as being equal to , where is the eccentricity of node (i.e., the maximum distance from to any other node) and 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 is the number of edge traversals made by the agent in excess of . 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 with a (small) penalty in . 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 with a linear penalty in 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.

Paper Structure

This paper contains 13 sections, 11 theorems, 5 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Model and Problem Definition
  4. Our Results
  5. Related Work
  6. Preliminaries
  7. Impossibility Result for Distance-constrained Exploration
  8. Overview of the Proof
  9. In a Nutshell
  10. Under the Hood
  11. Formal Proof
  12. Impossibility Result for Fuel-constrained Exploration
  13. Conclusion

Key Result

Lemma 2.1

For every positive integer $k$, and for every integer $n\geq k$, there exists a $k$-regular bipartite graph whose bipartition $\{U,U'\}$ satisfies $|U|=|U'|=n$.

Figures (2)

  • Figure 1: Example of a graph in $\mathcal{F}(4,8,7)$.
  • Figure 2: Illustrations of operations ${\tt switch\hbox{-}edges}$ and ${\tt move\hbox{-}gadget}$.

Theorems & Definitions (20)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Claim 3.1
  • Claim 3.2
  • Lemma 3.3
  • Claim 3.3
  • ...and 10 more