Polynomial Time Learning-Augmented Algorithms for NP-hard Permutation Problems
Evripidis Bampis, Bruno Escoffier, Dimitris Fotakis, Panagiotis Patsilinakos, Michalis Xefteris
TL;DR
This work develops a learning-augmented framework for NP-hard permutation problems by leveraging noisy pairwise predictions that are correct with probability at least $\tfrac{1}{2}+\varepsilon$. It identifies two structural properties—decomposition and $c$-locality—that permit exact polynomial-time solutions with high probability when augmented by $O(n \log n)$ predictions and a $O(\log n)$-position enhancement, via dynamic programming. The authors demonstrate this approach on classic NP-hard permutation problems (e.g., Maximum Acyclic Subgraph, Minimum Linear Arrangement, and scheduling variants) and show that the framework supports parsimonious use of predictions. They also establish hardness results: the $O(\log n)$ bound is tight for some problems, and stronger enhancements ($f(n)\log n$ with $f$ unbounded) render several problems intractable under ETH, highlighting fundamental limits. Overall, the paper provides a principled route to fast, prediction-informed exact solutions for a broad class of permutation-based NP-hard problems, while clarifying the exact thresholds where such gains cease to hold.
Abstract
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 $1/2+ε$. Moreover, this can be achieved with a parsimonious access to the predictions.