Tight upper and lower bounds for the quadratic knapsack problem through binary decision diagrams
研究利用二元决策图计算二次背包问题的紧上下界，提出结合对偶边界收紧的广度优先分支定界方法，在多个实例上优于现有精确算法，将平均对偶间隙降低最多10%。
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.