A logarithmic approximation for unsplittable flow on line graphs
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Bibliographic record
Abstract
We consider the unsplittable flow problem on a line. In this problem, we are given a set of n tasks, each specified by a start time s i , an end time t i , a demand d i > 0, and a profit p i > 0. A task, if accepted, requires d i units of “bandwidth” from time s i to t i and accrues a profit of p i . For every time t , we are also specified the available bandwidth c t , and the goal is to find a subset of tasks with maximum profit subject to the bandwidth constraints. We present the first polynomial time O (log n ) approximation algorithm for this problem. This significantly advances the state of the art, as no polynomial time o ( n ) approximation was known previously. Previous results for this problem were known only in more restrictive settings; in particular, either the instance satisfies the so-called “no-bottleneck” assumption: max i d i ≤ min t c t , or the ratio of both maximum to minimum demands and maximum to minimum capacities are polynomially (or quasi-polynomially) bounded in n . Our result, on the other hand, does not require these assumptions. Our algorithm is based on a combination of dynamic programming and rounding a natural linear programming relaxation for the problem. While there is an Ω( n ) integrality gap known for this LP relaxation, our key idea is to exploit certain structural properties of the problem to show that instances that are bad for the LP can in fact be handled using dynamic programming.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| 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