Approximation Schemes for Orienteering and Deadline TSP in Doubling Metrics
Bibliographic record
Abstract
In this paper we look at $k$-stroll, point-to-point orienteering, as well as the deadline TSP problem on graphs with bounded doubling dimension and bounded treewidth and present approximation schemes for them. Given a weighted graph $G=(V,E)$, start node $s\in V$, distances $d:E\rightarrow \mathbb{Q}^+$ and integer $k$. In the $k$-stroll problem the goal is to find a path starting at $s$ of minimum length that visits at least $k$ vertices. The dual problem to $k$-stroll is the rooted orienteering in which instead of $k$ we are given a budget $B$ and the goal is to find a walk of length at most $B$ starting at $s$ that visits as many vertices as possible. In the P2P orienteering we are given start and end nodes $s,t$ for the path. In the deadline TSP we are given a deadline $D(v)$ for each $v\in V$ and the goal is to find a walk starting at $s$ that visits as many vertices as possible before their deadline. The best approximation for rooted or P2P orienteering is $(2+ε)$-approximation [12] and $O(\log n)$-approximation for deadline TSP [3]. There is no known approximation scheme for deadline TSP for any metric (not even trees). Our main result is the first approximation scheme for deadline TSP on metrics with bounded doubling dimension. To do so we first show if $G$ is a metric with doubling dimension $κ$ and aspect ratio $Δ$, there is a $(1+ε)$-approximation that runs in time $n^{O\left(\left(\logΔ/ε\right)^{2κ+1}\right)}$. We then extend these to obtain an approximation scheme for deadline TSP when the distances and deadlines are integer which runs in time $n^{O\left(\left(\log Δ/ε\right)^{2κ+2}\right)}$. For graphs with treewidth $ω$ we show how to solve $k$-stroll and P2P orienteering exactly in polynomial time and a $(1+ε)$-approximation for deadline TSP in time $n^{O((ω\logΔ/ε)^2)}$.
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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 | 0.013 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.011 |
| Open science | 0.005 | 0.004 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.011 | 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".