Approximating Min-Cost Chain-Constrained Spanning Trees: A Reduction from Weighted to Unweighted Problems
Bibliographic record
Abstract
We study the {\em min-cost chain-constrained spanning-tree} (abbreviated \mcst) problem: find a min-cost spanning tree in a graph subject to degree constraints on a nested family of node sets. We devise the {\em first} polytime algorithm that finds a spanning tree that (i) violates the degree constraints by at most a constant factor {\em and} (ii) whose cost is within a constant factor of the optimum. Previously, only an algorithm for {\em unweighted} \cst was known \cite{olver}, which satisfied (i) but did not yield any cost bounds. This also yields the first result that obtains an $O(1)$-factor for {\em both} the cost approximation and violation of degree constraints for any spanning-tree problem with general degree bounds on node sets, where an edge participates in a super-constant number of degree constraints. A notable feature of our algorithm is that we {\em reduce} \mcst to unweighted \cst (and then utilize \cite{olver}) via a novel application of {\em Lagrangian duality} to simplify the {\em cost structure} of the underlying problem and obtain a decomposition into certain uniform-cost subproblems. We show that this Lagrangian-relaxation based idea is in fact applicable more generally and, for any cost-minimization problem with packing side-constraints, yields a reduction from the weighted to the unweighted problem. We believe that this reduction is of independent interest. As another application of our technique, we consider the {\em $k$-budgeted matroid basis} problem, where we build upon a recent rounding algorithm of \cite{BansalN16} to obtain an improved $n^{O(k^{1.5}/ε)}$-time algorithm that returns a solution that satisfies (any) one of the budget constraints exactly and incurs a $(1+ε)$-violation of the other budget constraints.
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How this classification was reachedexpand
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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".