Online Ordinal Problems: Optimality of Comparison-based Algorithms and their Cardinal Complexity

We consider ordinal online problems, i.e., tasks that only require pairwise comparisons between elements of the input. A classic example is the secretary problem and the game of googol, as well as its multiple combinatorial extensions such as $(J,K)$-secretary, $2$-sided game of googol, ordinal-competitive matroid secretary. A natural approach to these tasks is to use ordinal algorithms that at each step only consider relative ranking among the arrived elements, without looking at the numerical values of the input. We formally study the question of how cardinal algorithms can improve upon ordinal algorithms. We give first a universal construction of the input distribution for any ordinal online problem, such that the advantage of any cardinal algorithm over the ordinal algorithms is at most $1+\varepsilon$ for arbitrary small $\varepsilon> 0$. As an implication, previous lower bounds for the aforementioned variants of secretary problems hold not only against ordinal algorithms, but also against any online algorithm. However, the value range of the input elements in our construction is huge: $N=O\left(\frac{n^3\cdot n!\cdot n!}{\varepsilon}\right)\uparrow\uparrow(n-1)$ (tower of exponents) for an input sequence of length $n$. As a second result, we identify a class of natural ordinal problems and find cardinal algorithm with a matching advantage of $1+ \Omega \left(\frac{1}{\log^{{(c)}N}\right),$} where $\log^{{(c)}N=\log\ldots\log} N$ with $c$ iterative logs and $c$ is an arbitrary constant. Further, we introduce the cardinal complexity for any given ordinal online task: the minimum size $N(\varepsilon)$ of different numerical values in the input such the advantage of cardinal over ordinal algorithms is at most $1+\varepsilon$. As a third result, we show that the game of googol has much lower cardinal complexity of $N=O\left(\left(\frac{n}{\varepsilon}\right)^n\right)$.