Having a mixed-integer linear program (MIP) or a nonlinear program (NLP) in the second stage further aggravates the intractability, even when specialized algorithms that exploit problem structure are employed.

We start with describing the pseudocode of our algorithm.Algorithm 1: Algorithm to compute closest weak fair ranking under Kendall tauInput: Input ranking π S Initialize a set P . Iterate over ranking σ, and count the fraction of elements in the top-k from each group. We first provide the pseudocode of our algorithm.Algorithm 2: Algorithm to compute closest fair ranking under Kendall tauInput: Input ranking π S If not, return "No fair ranking exists"; else continue. The claim now follows.Claim 3.9. From Claim 3.7 we know that Algorithm 1 and the optimal solution have the same set of elements.

This paper provides the first polynomial time guarantees for the Burer-Monteiro method.

We thank all the reviewers for their feedback and pointers to relevant papers. This includes (Kendall et al., 2018), where they learn Kendall et al. 2018), we consider different loss functions on the same output space. There are specific reasons we did not use several multi-task learning algorithms mentioned by REV4 as baselines. Kendall et al. (2018) assumes that all base losses are applications of the same function (max likelihood in this case) We don't see how this method can be extended to our scenario where base losses do not necessarily Moreover, our regularization admits a very different nature. However, directly normalizing the base losses was sufficient for our experiments.

Our approach reduces the problem into a sequence of plug-in classifier learning tasks.