Online Exploration of Rectangular Grids

TL;DR

This paper studies the problem of exploring unknown rectangular grid environments with autonomous agents, providing bounds on the efficiency of exploration strategies and analyzing the complexity in different graph settings.

Contribution

It introduces a new model for exploration with partial knowledge, establishes lower bounds, and presents competitive algorithms for undirected grid exploration.

Findings

Lower bound of 11/9 on competitive ratio for ladder graphs

A 4-competitive algorithm for undirected grid exploration

Greedy strategies have linear lower bounds in directed grid exploration

Abstract

In this paper, we consider the problem of exploring unknown environments with autonomous agents. We model the environment as a graph with edge weights and analyze the task of visiting all vertices of the graph at least once. The hardness of this task heavily depends on the knowledge and the capabilities of the agent. In our model, the agent sees the whole graph in advance, but does not know the weights of the edges. As soon as it arrives in a vertex, it can see the weights of all the outgoing edges. We consider the special case of two different edge weights $1$ and $k$ and prove that the problem remains hard even in this case. We prove a lower bound of $11/9$ on the competitive ratio of any deterministic strategy for exploring a ladder graph and complement this result by a $4$-competitive algorithm. All of these results hold for undirected graphs. Exploring directed graphs, where the direction of the edges is not known beforehand, seems to be much harder. Here, we prove that a natural greedy strategy has a linear lower bound on the competitive ratio both in ladders and square grids.