Partial-order analogue of the secretary problem the binary tree case

The best choice problem for posets; colored complete binary trees

“…In Morayne (1998) it was proved that for chains of length t, where 2t ≤n, playing optimally we do not stop; actually, the justification is similar as for the previous case (using τ S ).…”

“…In Morayne (1998) it is proved that there are more good than bad chains of length t in CBTn when 2t >n. This justifies the optimality of stopping in the first case.…”

“…Optimal algorithms for simple non-linear posets were found in Kaźmierczak (2013), Tkocz and Tkocz and Kaźmierczak. A poset secretary problem was also considered in Garrod et al (2012) where every candidate has an equally qualified twin. An optimal algorithm for the posets whose Hasse diagrams are complete binary trees of given length was found in Morayne (1998). Additional assumptions about a model are also possible in the poset version of the secretary problem.…”

“…Posets, e.g., have sides and it is natural to assume that the selector can recognize from which side a particular element comes. In this paper we enrich the complete binary tree model considered in Morayne (1998). Namely, in the original version the selector in a given moment can see only the poset induced by the elements that have come so far, having no information from which side of the tree the observed elements came.…”

“…Morayne (1998) or Preater (1999 for further details. We hope the following description will be sufficient to follow the argument given and to enable the reader to add the formalism lacked.…”