We study the problem of exploring an unknown, strongly connected directed graph. Starting at some node of the graph, we must visit every edge and every node at least once. The goal is to minimize the number of edge traversals. It is known that the competitive ratio of online algorithms for this problem depends on the deficiency d of the graph, which is the minimum number of edges that must be added to make the graph Eulerian. We present the first deterministic online exploration algorithm whose competitive ratio is polynomial in d (it is O(d~8)).
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