The Cardinal Complexity of Online Ordinal Problems

We consider ordinal online problems, i.e., tasks that only depend on the pairwise comparisons between elements of the input. E.g., the secretary problem and the game of googol. The natural approach to these tasks is to use ordinal online 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 (that can use numerical values of the input) can improve upon ordinal algorithms. We give 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 $\varepsilon> 0$. However, the value range of the input elements in this construction is huge: $O\left(\frac{n^3\cdot n!}{\varepsilon}\right)\uparrow\uparrow (n-1)$ for an input sequence of length $n$. Second, we identify a natural family of core problems and find a cardinal algorithm with a matching advantage of $1+ \Omega \left(\frac{1}{\log^{(c)}N}\right),$ where $\log^{(c)}N=\log\log\ldots\log N$ with $c$ iterative logs and $c$ is an arbitrary constant $c\le n-2$. Third, we construct an input distribution of only exponential size $N=O((n/\varepsilon)^n)$ for the game of googol such that any cardinal algorithm has advantage of at most $1+\varepsilon$ over ordinal algorithms for arbitrary small $\varepsilon> 0$. Finally, we study the dependency on $n$ of the core problem. We provide an efficient construction of size $O(n)$, if we allow cardinal algorithms to have constant factor advantage against ordinal algorithms.