Improved Approximation Guarantees for Lower-Bounded Facility Location
Bibliographic record
Abstract
We consider the {\em lower-bounded facility location} (\lbfl) problem (also sometimes called {\em load-balanced facility location}), which is a generalization of {\em uncapacitated facility location} (\ufl), where each open facility is required to serve a certain {\em minimum} amount of demand. More formally, an instance $\I$ of \lbfl is specified by a set $\F$ of facilities with facility-opening costs $\{f_i\}$, a set $\D$ of clients, and connection costs $\{c_{ij}\}$ specifying the cost of assigning a client $j$ to a facility $i$, where the $c_{ij}$s form a metric. A feasible solution specifies a subset $F$ of facilities to open, and assigns each client $j$ to an open facility $i(j)\in F$ so that each open facility serves {\em at least $M$ clients}, where $M$ is an input parameter. The cost of such a solution is $\sum_{i\in F}f_i+\sum_j c_{i(j)j}$, and the goal is to find a feasible solution of minimum cost. The current best approximation ratio for \lbfl is 448 \cite{Svitkina08}. We substantially advance the state-of-the-art for \lbfl by devising an approximation algorithm for \lbfl that achieves a significantly-improved approximation guarantee of 82.6. Our improvement comes from a variety of ideas in algorithm design and analysis, which also yield new insights into \lbfl.
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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) | 0.004 | 0.001 |
| Meta-epidemiology (broad) | 0.004 | 0.003 |
| Bibliometrics | 0.002 | 0.005 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.004 | 0.010 |
| Open science | 0.007 | 0.005 |
| Research integrity | 0.003 | 0.007 |
| Insufficient payload (model declined to judge) | 0.020 | 0.005 |
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".