Parameterized Approximation for Capacitated $d$-Hitting Set with Hard Capacities
Bibliographic record
Abstract
The \textsc{Capacitated $d$-Hitting Set} problem involves a universe $U$ with a capacity function $\mathsf{cap}: U \rightarrow \mathbb{N}$ and a collection $\mathcal{A}$ of subsets of $U$, each of size at most $d$. The goal is to find a minimum subset $S \subseteq U$ and an assignment $ϕ: \mathcal{A} \rightarrow S$ such that for every $A \in \mathcal{A}$, $ϕ(A) \in A$, and for each $x \in U$, $|ϕ^{-1}(x)| \leq \mathsf{cap}(x)$. For $d=2$, this is known as \textsc{Capacitated Vertex Cover}. In the weighted variant, each element of $U$ has a positive integer weight, with the objective of finding a minimum-weight capacitated hitting set. Chuzhoy and Naor [SICOMP 2006] provided a factor-3 approximation for \textsc{Capacitated Vertex Cover} and showed that the weighted case lacks an $o(\log n)$-approximation unless $P=NP$. Kao and Wong [SODA 2017] later independently achieved a $d$-approximation for \textsc{Capacitated $d$-Hitting Set}, with no $d - ε$ improvements possible under the Unique Games Conjecture. Our main result is a parameterized approximation algorithm with runtime $\left(\frac{k}ε\right)^k 2^{k^{O(kd)}}(|U|+|\mathcal{A}|)^{O(1)}$ that either concludes no solution of size $\leq k$ exists or finds $S$ of size $\leq 4/3 \cdot k$ and weight at most $2+ε$ times the minimum weight for solutions of size $\leq k$. We further show that no FPT-approximation with factor $c > 1$ exists for unweighted \textsc{Capacitated $d$-Hitting Set} with $d \geq 3$, nor with factor $2 - ε$ for the weighted version, assuming the Exponential Time Hypothesis. These results extend to \textsc{Capacitated Vertex Cover} in multigraphs. Additionally, a variant of multi-dimensional \textsc{Knapsack} is shown hard to FPT-approximate within $2 - ε$.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.014 |
| Meta-epidemiology (narrow) | 0.005 | 0.002 |
| Meta-epidemiology (broad) | 0.004 | 0.003 |
| Bibliometrics | 0.002 | 0.007 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.005 | 0.008 |
| Open science | 0.009 | 0.005 |
| Research integrity | 0.004 | 0.006 |
| Insufficient payload (model declined to judge) | 0.025 | 0.004 |
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".