Title: Combinatorial Secretary Problems with Ordinal Information,
Abstract: The secretary problem is a classic model for online decision making.
Recently, combinatorial extensions such as matroid or matching secretary
problems have become an important tool to study algorithmic problems in dynamic
markets. Here the decision maker must know the numerical value of each arriving
element, which can be a demanding informational assumption. In this paper, we
initiate the study of combinatorial secretary problems with ordinal
information, in which the decision maker only needs to be aware of a preference
order consistent with the values of arrived elements. The goal is to design
online algorithms with small competitive ratios.
For a variety of combinatorial problems, such as bipartite matching, general
packing LPs, and independent set with bounded local independence number, we
design new algorithms that obtain constant competitive ratios.
For the matroid secretary problem, we observe that many existing algorithms
for special matroid structures maintain their competitive ratios even in the
ordinal model. In these cases, the restriction to ordinal information does not
represent any additional obstacle. Moreover, we show that ordinal variants of
the submodular matroid secretary problems can be solved using algorithms for
the linear versions by extending [Feldman and Zenklusen, 2015]. In contrast, we
provide a lower bound of $\Omega(\sqrt{n}/(\log n))$ for algorithms that are
oblivious to the matroid structure, where $n$ is the total number of elements.
This contrasts an upper bound of $O(\log n)$ in the cardinal model, and it
shows that the technique of thresholding is not sufficient for good algorithms
in the ordinal model.