Improved Approximation Algorithms for Capacitated Network Design and Flexible Graph Connectivity
Bibliographic record
Abstract
We present improved approximation algorithms for some problems in the related areas of Capacitated Network Design and Flexible Graph Connectivity. In the Cap-k-ECSS problem, we are given a graph G = (V,E) whose edges have non-negative costs and positive integer capacities, and the goal is to find a minimum-cost edge-set F such that every non-trivial cut of the graph G' = (V,F) has capacity at least k. Let n = |V| and let u_{min} (respectively, u_{max}) denote the minimum (respectively, maximum) capacity of an edge; assume that u_{max} ≤ k. We present an O(log({k}/u_{min}))-approximation algorithm for the Cap-k-ECSS problem, asymptotically improving upon the previous best approximation ratio of min(O(log{n}), k, 2u_{max}, 6 ⋅ {⌈ k/u_{min} ⌉}) whenever log(k/u_{min}) = o(log{n}) and u_{max} is sufficiently large. In the (p,q)-Flexible Graph Connectivity problem, denoted (p,q)-FGC, the input is a graph G = (V, E) where E is partitioned into safe and unsafe edges, and the goal is to find a minimum-cost edge-set F such that the subgraph G' = (V, F) remains p-edge connected upon removal of any q unsafe edges from F. We present an 8-approximation algorithm for the (1,q)-FGC problem that improves upon the previous best approximation ratio of (q+1). Both of our results are obtained by using natural LP relaxations strengthened with the knapsack-cover inequalities, and then, during the rounding process, utilizing a recent O(1)-approximation algorithm for the Cover Small Cuts problem. In the latter problem, the goal is to find a minimum-cost set of links such that each non-trivial cut of capacity less than a specified value is covered by a link. We also show that the problem of covering small cuts inherently arises in another variant of (p,q)-FGC. Specifically, we give Cook reductions that preserve approximation ratios within O(1) factors between the (2,q)-FGC problem and the 2-Cover Small Cuts problem; in the latter problem, each small cut needs to be covered by two links.
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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 | 0.002 | 0.010 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.004 | 0.002 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.006 | 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".