  In this work we consider the problem of fitting Random Utility Models (RUMs)
to user choices. Given the winner distributions of the subsets of size $k$ of a
universe, we obtain a polynomial-time algorithm that finds the RUM that best
approximates the given distribution on average. Our algorithm is based on a
linear program that we solve using the ellipsoid method. Given that its
corresponding separation oracle problem is NP-hard, we devise an approximate
separation oracle that can be viewed as a generalization of the weighted
feedback arc set problem to hypergraphs. Our theoretical result can also be
made practical: we obtain a heuristic that is effective and scales to
real-world datasets.
