Secretary Matching Meets Probing with Commitment
Bibliographic record
Abstract
We consider the online bipartite matching problem within the context of stochastic probing with commitment. This is the one-sided online bipartite matching problem where edges adjacent to an online node must be probed to determine if they exist based on edge probabilities that become known when an online vertex arrives. If a probed edge exists, it must be used in the matching. We consider the competitiveness of online algorithms in the adversarial order model (AOM) and the secretary/random order model (ROM). More specifically, we consider an unknown bipartite stochastic graph G = (U,V,E) where U is the known set of offline vertices, V is the set of online vertices, G has edge probabilities (p_{e})_{e ∈ E}, and G has edge weights (w_{e})_{e ∈ E} or vertex weights (w_u)_{u ∈ U}. Additionally, G has a downward-closed set of probing constraints (𝒞_{v})_{v ∈ V}, where 𝒞_v indicates which sequences of edges adjacent to an online vertex v can be probed. This model generalizes the various settings of the classical bipartite matching problem (i.e. with and without probing). Our contributions include the introduction and analysis of probing within the random order model, and our generalization of probing constraints which includes budget (i.e. knapsack) constraints. Our algorithms run in polynomial time assuming access to a membership oracle for each 𝒞_v. In the vertex weighted setting, for adversarial order arrivals, we generalize the known 1/2 competitive ratio to our setting of 𝒞_v constraints. For random order arrivals, we show that the same algorithm attains an asymptotic competitive ratio of 1-1/e, provided the edge probabilities vanish to 0 sufficiently fast. We also obtain a strict competitive ratio for non-vanishing edge probabilities when the probing constraints are sufficiently simple. For example, if each 𝒞_v corresponds to a patience constraint 𝓁_v (i.e., 𝓁_v is the maximum number of probes of edges adjacent to v), and any one of following three conditions is satisfied (each studied in previous papers), then there is a conceptually simple greedy algorithm whose competitive ratio is 1-1/e. - When the offline vertices are unweighted. - When the online vertex probabilities are "vertex uniform"; i.e., p_{u,v} = p_v for all (u,v) ∈ E. - When the patience constraint 𝓁_v satisfies 𝓁_v ∈ {[1,|U|} for every online vertex; i.e., every online vertex either has unit or full patience. Finally, in the edge weighted case, we match the known optimal 1/e asymptotic competitive ratio for the classic (i.e. without probing) secretary matching problem.
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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 | 0.019 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.004 | 0.008 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.003 | 0.004 |
| Insufficient payload (model declined to judge) | 0.009 | 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".