Optimal constrained graph exploration

Abstract

We address the problem of constrained exploration of an unknown graph G = ( V , E ) from a given start node s with either a tethered robot or a robot with a fuel tank of limited capacity, the former being a tighter constraint. In both variations of the problem, the robot can only move along the edges of the graph, for example, it cannot jump between nonadjacent nodes. In the tethered robot case, if the tether (rope) has length l , then the robot must remain within distance l from the start node s . In the second variation, a fuel tank of limited capacity forces the robot to return to s after traversing C edges. The efficiency of algorithms for both variations of the problem is measured by the number of edges traversed during the exploration. We present an algorithm for a tethered robot that explores the graph in Θ(| E |) edge traversals. The problem of exploration using a robot with a limited fuel tank capacity can be solved with a simple reduction from the tethered robot case and also yields a Θ(| E |) algorithm. This improves on the previous best-known bound of O (| E | + | V |log 2 | V |). Since the lower bound for the graph exploration problems is Ω(| E |), our algorithm is optimal within a constant factor.