Polynomial Time Learning Augmented Algorithms for NP-hard Permutation Problems

We consider a learning augmented framework for NP-hard permutation problems. The algorithm has access to predictions telling, given a pair $u,v$ of elements, whether $u$ is before $v$ or not in an optimal solution. Building on the work of Braverman and Mossel (SODA 2008), we show that for a class of optimization problems including scheduling, network design and other graph permutation problems, these predictions allow to solve them in polynomial time with high probability, provided that predictions are true with probability at least $1/2+\epsilon$. Moreover, this can be achieved with a parsimonious access to the predictions.

Many important tasks in computing, such as scheduling jobs or designing efficient networks, require finding the best way to arrange a list of items. Unfortunately, solving such ordering problems is often extremely hard, even for powerful computers.In this work, we show how predictions from a machine learning model can make these problems much easier to solve. The idea is simple: the model gives us a guess, for any two items, about which one should come first in a best-possible solution. Even if these guesses are only slightly better than random (just a bit more than 50% accurate), we can use them to find the best-possible solution efficiently.The key insight is that we don’t need perfect predictions, just a slightly helpful advice. We also need to ask the model about very few pairs, which keeps the process efficient. This approach combines the strengths of algorithms and machine learning to tackle problems that are otherwise out of reach.