A Logarithmic Integrality Gap for Generalizations of Quasi-Bipartite Instances of Directed Steiner Tree
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Bibliographic record
Abstract
In the classic Directed Steiner Tree problem (DST), we are given an edge-weighted directed graph G = (V,E) with n nodes, a specified root node r ∈ V, and k terminals X ⊆ V-{r}. The goal is to find the cheapest F ⊆ E such that r can reach any terminal using only edges in F. Designing approximation algorithms for DST is quite challenging, to date the best approximation guarantee of a polynomial-time algorithm for DST is O(k^ε) for any constant ε > 0 [Charikar et al., 1999]. For network design problems like DST, one often relies on natural cut-based linear programming (LP) relaxations to design approximation algorithms. In general, the integrality gap of such an LP for DST is known to have a polynomial integrality gap lower bound [Zosin and Khuller, 2002; Li and Laekhanukit, 2021]. So particular interest has been invested in special cases or in strengthenings of this LP. In this work, we show the integrality gap is only O(log k) for instances of DST where no Steiner node has both an edge from another Steiner node and an edge to another Steiner node, i.e. the longest path using only Steiner nodes has length at most 1. This generalizes the well-studied case of quasi-bipartite DST where no edge has both endpoints being Steiner nodes. Our result is also optimal in the sense that the integrality gap can be as bad as poly(n) even if the longest path with only Steiner nodes has length 2.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it