Do Language Models Know Which Constraints Belong Together?

Junbo Jacob Lian, Kaixuan Chen, Hanzhang Qin, Huiling Chen, Chaoyu Zhang +1

Dantzig-Wolfe reformulation strengthens the linear relaxation of a mixed-integer program by convexifying a chosen set of constraint families, and every candidate costs a column generation run. The choice is hard when families strengthen the bound only together, because a search that adds one family at a time then stops at the linear relaxation. The classical inequalities of lot sizing, network design and facility location rest on such zero-marginal groups. We prove that when each evaluation convexifies at most $s$ of $k$ families, any method that observes only bounds needs a covering number of evaluations, of order $k^w$, to find a hidden group of $w$ families, also on actual Dantzig-Wolfe relaxations. A prediction that misses $e$ members reduces the cost to $Θ(k^e)$, through one staged search that does not need to know $e$, so the evaluations a prediction saves measure what the predictor knows. We study two sources of predictions. A one-pass lookahead at an optimal LP dual is provably blind to groups whose families share no variable, and a language model reads the names and text of the model. JointDW verifies the prediction of the language model with the staged search and…