Lower Bounds for Graph Exploration Using Local Policies

TL;DR

This paper establishes lower bounds for local graph exploration strategies, showing that node-based policies can be exponentially inefficient, contrasting with edge-based strategies that have polynomial bounds.

Contribution

It introduces and analyzes lower bounds for node-based local exploration policies, highlighting their potential inefficiency compared to edge-based strategies.

Findings

Node-based strategies can have superpolynomial exploration time.

Edge-based strategies can achieve polynomial exploration time.

Node counters are more natural but less efficient for exploration.

Abstract

We give lower bounds for various natural node- and edge-based local strategies for exploring a graph. We consider this problem both in the setting of an arbitrary graph as well as the abstraction of a geometric exploration of a space by a robot, both of which have been extensively studied. We consider local exploration policies that use time-of-last- visit or alternatively least-frequently-visited local greedy strategies to select the next step in the exploration path. Both of these strategies were previously considered by Cooper et al. (2011) for a scenario in which counters for the last visit or visit frequency are attached to the edges. In this work we consider the case in which the counters are associated with the nodes, which for the case of dual graphs of geometric spaces could be argued to be intuitively more natural and likely more efficient. Surprisingly, these alternate strategies give worst-case superpolynomial/ exponential time for exploration, whereas the least-frequently visited strategy for edges has a polynomially bounded exploration time, as shown by Cooper et al. (2011).