# Combinatorial Secretary Problems with Ordinal Information

## TL;DR

This paper explores combinatorial secretary problems with only ordinal information, developing algorithms with constant competitive ratios for various problems and establishing bounds that highlight the challenges of the ordinal setting.

## Contribution

It introduces the ordinal model for combinatorial secretary problems, designs new algorithms with constant competitive ratios, and provides lower bounds illustrating the limitations of oblivious algorithms.

## Key findings

- Algorithms achieve constant competitive ratios for bipartite matching, packing LPs, and independent sets.
- Many existing algorithms for special matroids perform well in the ordinal model.
- Lower bounds show the difficulty of designing oblivious algorithms in the ordinal setting.

## 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.