A logarithmic approximation for unsplittable flow on line graphs
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 si, an end time ti, a demand di > 0, and a profit pi > 0. A task, if accepted, requires di units of “bandwidth” from time si to ti and accrues a profit of pi. For every time t, we are also specified the available bandwidth ct, and the goal is to find a subset of tasks with maximum profit subject to the bandwidth constraints. In this paper, we present the first polynomial-time O(log n)-approximation algorithm for this problem. No polynomial-time o(n)-approximation was known prior to this work. Previous results for this problem were known only in more restrictive settings, in particular, either if the given instance satisfies the so-called “no-bottleneck” assumption: maxi di ≤ mint ct, or else if the ratio of the maximum to the minimum demands and ratio of the maximum to the minimum capacities are polynomially (or quasi-polynomially) bounded in n. Our result, on the other hand, does not require any of 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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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 | 0.006 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.010 | 0.002 |
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".