Optimal graph exploration without good maps

A robot has to visit all nodes and traverse all edges of an unknown undirected connected graph, using as few edge traversals as possible. The quality of an exploration algorithm A is measured by comparing its cost (number of edge traversals) to that of the optimal algorithm having full knowledge of the graph. The ratio between these costs, maximized over all starting nodes in the graph and over all graphs in a given class U, is called the overhead of algorithm A for the class U of graphs. We construct natural exploration algorithms, for various classes of graphs, that have smallest, or - in one case - close to smallest, overhead. An important contribution of this paper is establishing lower bounds that prove optimality of these exploration algorithms.