Approximation Algorithms for Correlated Knapsack Orienteering
Bibliographic record
Abstract
We consider the {\em correlated knapsack orienteering} (CSKO) problem: we are given a travel budget $B$, processing-time budget $W$, finite metric space $(V,d)$ with root $ρ\in V$, where each vertex is associated with a job with possibly correlated random size and random reward that become known only when the job completes. Random variables are independent across different vertices. The goal is to compute a $ρ$-rooted path of length at most $B$, in a possibly adaptive fashion, that maximizes the reward collected from jobs that are processed by time $W$. To our knowledge, CSKO has not been considered before, though prior work has considered the uncorrelated problem, {\em stochastic knapsack orienteering}, and {\em correlated orienteering}, which features only one budget constraint on the {\em sum} of travel-time and processing-times. We show that the {\em adaptivity gap of CSKO is not a constant, and is at least $Ω\bigl(\max\sqrt{\log{B}},\sqrt{\log\log{W}}\}\bigr)$}. Complementing this, we devise {\em non-adaptive} algorithms that obtain: (a) $O(\log\log W)$-approximation in quasi-polytime; and (b) $O(\log W)$-approximation in polytime. We obtain similar guarantees for CSKO with cancellations, wherein a job can be cancelled before its completion time, foregoing its reward. We also consider the special case of CSKO, wherein job sizes are weighted Bernoulli distributions, and more generally where the distributions are supported on at most two points (2-CSKO). Although weighted Bernoulli distributions suffice to yield an $Ω(\sqrt{\log\log B})$ adaptivity-gap lower bound for (uncorrelated) {\em stochastic orienteering}, we show that they are easy instances for CSKO. We develop non-adaptive algorithms that achieve $O(1)$-approximation in polytime for weighted Bernoulli distributions, and in $(n+\log B)^{O(\log W)}$-time for the more general case of 2-CSKO.
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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 | 0.008 |
| Meta-epidemiology (narrow) | 0.003 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.005 | 0.003 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.008 | 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".