Lehman's Theorem and the Directed Steiner Tree Problem
Bibliographic record
Abstract
In the directed Steiner tree problem, we are given a digraph, nonnegative arc weights, a subset of vertices called terminals, and a special terminal called the root. The goal is to compute a minimum weight directed tree that connects each terminal to the root. We study the classical directed cut linear programming (LP) formulation which has a variable for every arc, and a constraint for every cut that separates a terminal from the root. For what instances is the directed cut LP integral? In this paper we demonstrate how the celebrated theorem of Lehman [Math. Program., 17 (1979), pp. 403--417] on minimally nonideal clutters provides a framework for deriving answers to this question. Specifically, we show that this framework yields short proofs of the optimum arborescences theorem and the integrality result for series-parallel digraphs. Furthermore, we use this framework to show that the directed cut linear program is integral for digraphs that are acyclic and have at most two nonterminal vertices.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.004 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".