Asymptotic Quasi-Polynomial Time Approximation Scheme for Resource Minimization for Fire Containment
Bibliographic record
Abstract
Resource Minimization Fire Containment (RMFC) is a natural model for optimal inhibition of harmful spreading phenomena on a graph. In the RMFC problem on trees, we are given an undirected tree G, and a vertex r where the fire starts at, called root. At each time step, the firefighters can protect up to B vertices of the graph while the fire spreads from burning vertices to all their neighbors that have not been protected so far. The task is to find the smallest B that allows for saving all the leaves of the tree. The problem is hard to approximate up to any factor better than 2 even on trees unless P = NP (King and MacGillivray in Discret Math 310(3):614–621, 2010). Chalermsook and Chuzhoy (In: Proceedings of the 21st annual ACM-SIAM symposium on discrete algorithms, SODA 2010, Austin, Texas, USA, 17–19 Jan 2010, SIAM, pp 1334–1349, 2010) presented a Linear Programming (LP) based $$O(\log ^* n)$$ approximation for RMFC on trees that matches the integrality gap of the natural Linear Programming relaxation. This was recently improved by Adjiashvili et al. (ACM Trans Algorithms 15(2):20:1–20:33, 2019) to a 12-approximation through a combination of LP rounding along with several new techniques. In this paper we present an asymptotic QPTAS for RMFC on trees. More specifically, let $$\epsilon >0$$ , and $$\mathcal {I}$$ be an instance of RMFC where the optimum number of firefighters to save all the leaves is $$OPT(\mathcal {I})$$ . We present an algorithm which uses at most $$\lceil (1+\epsilon )OPT(\mathcal {I})\rceil $$ many firefighters at each time step and runs in time $$n^{O(\log \log n/\epsilon )}$$ . This suggests that the existence of an asymptotic PTAS is plausible especially since the exponent is $$O(\log \log n)$$ , not $$O(\log n)$$ . Our result combines a more refined height reduction lemma than the one in Adjiashvili et al. (2019) with LP rounding and dynamic programming to find the solution. We also apply our height reduction lemma to the algorithm provided in Adjiashvili et al. (2019) plus a more careful analysis to improve their 12-approximation and provide a polynomial time ( $$5+\epsilon $$ )-approximation.
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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 | 0.003 | 0.013 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.005 | 0.008 |
| Open science | 0.007 | 0.005 |
| Research integrity | 0.003 | 0.006 |
| Insufficient payload (model declined to judge) | 0.016 | 0.003 |
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".