On the Linear Ordering Problem and the Rankability of Data

Abstract

In 2019, Anderson et al. proposed the concept of rankability, which refers to a dataset's inherent ability to be meaningfully ranked. In this article, we give an expository review of the linear ordering problem (LOP) and then use it to analyze the rankability of data. Specifically, the degree of linearity is used to quantify what percentage of the data aligns with an optimal ranking. In a sports context, this is analogous to the number of games that a ranking can correctly predict in hindsight. In fact, under the appropriate objective function, we show that the optimal rankings computed via the LOP maximize the hindsight accuracy of a ranking. Moreover, we develop a binary program to compute the maximal Kendall tau ranking distance between two optimal rankings, which can be used to measure the diversity among optimal rankings without having to enumerate all optima. Finally, we provide several examples from the world of sports and college rankings to illustrate these concepts and demonstrate our results.

Neural Combinatorial Optimization: a New Player in the Field

“…Particularly, the LOP is a permutation problem that, in 1979, was proven to be NP-hard by Garey and Johnson. Since then, and due to its applicability in fields such as machine translation [35], economics [36], corruption perception [37] and rankings in sports or other tournaments [38], [39], the LOP has gained popularity and it is easy to find a wide variety of works that have dealt with it [40].…”