The best choice problem for upward directed graphs

We consider a generalization of the best choice problem to upward directed graphs. We describe a strategy for choosing a maximal element (i.e., an element with no outgoing edges) when a selector knows in advance only the number n of vertices of the graph. We show that, as long as the number of elements dominated directly by the maximal ones is not greater than c(1)root n for some positive constant c(1) and the indegree of remaining vertices is bounded by a constant D, the probability p(n) of the right choice according to our strategy satisfies lim inf(n ->infinity) p(n) root n >= delta > 0, where delta is a constant depending on c(1) and D. (c) 2012 Elsevier B.V. All rights reserved.