Tight upper and lower bounds for the quadratic knapsack problem through binary decision diagrams
Bibliographic record
Abstract
The Quadratic Knapsack Problem (QKP) is a challenging combinatorial optimization problem that has attracted significant attention due to its complexity and practical applications. In recent years, Binary Decision Diagrams (BDDs) have emerged as a powerful tool in combinatorial optimization, providing efficient bounds. In the literature of the QKP, all the exact methods are based on computing tight bounds before applying branch-and-bound (B&B) schemes. We advance this literature in this work by leveraging BDDs to compute bounds more effectively. We propose a novel integration of dual-bound tightening within a BDD-based B&B framework, employing a Breadth-First Search (BFS) strategy. Our approach addresses the critical limitation of existing BDD-based B&B methods, which often lack robust dual-bound tightening mechanisms. Furthermore, we propose several efficient compilation techniques of BDDs for the QKP. Through extensive experimentation on several categories of QKP instances, we demonstrate that our method competes and often surpasses the bounding stages of the leading exact algorithms. Notably, our approach reduces the average duality gap by up to 10% for the class of Hidden Clique QKP instances, showcasing its potential. Furthermore, our findings indicate that the BFS B&B method outperforms state-of-the-art BDD B&B approaches across all tested QKP instances, highlighting its effectiveness and potential for broader application.
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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 | 0.004 | 0.021 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.004 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.009 | 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".