The Quadratic Knapsack Problem (QKP) is a well-known $\mathcal{NP}$-hard combinatorial optimisation problem, with many practical applications. We present a ‘cut-and-branch’ algorithm for the QKP, in which a cutting-plane phase is followed by a branch-and-bound phase. The cutting-plane phase is more sophisticated than the existing ones in the literature, incorporating several classes of cutting planes, two primal heuristics, and several rules for eliminating variables and constraints. Computational results show that the algorithm is competitive.

中文翻译：

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二次背包问题的割分算法

所述二次背包问题（QKP）是一种公知的$\mathcal{NP}$困难的组合优化问题，具有许多实际应用。我们提出了一种针对QKP的“剪切和分支”算法，其中在剪切平面阶段之后是分支定界阶段。切割平面阶段比文献中的现有阶段更为复杂，合并了几类切割平面，两种原始启发式方法以及一些用于消除变量和约束的规则。计算结果表明该算法具有竞争力。