Abstract
We study the classical quadratic knapsack problem (QKP) on special graph classes. In this case the quadratic terms of the objective function are present only for certain pairs of knapsack items. These pairs are represented by the edges of a graph G=(V,E) whose vertices represent the knapsack items. We show that QKP permits an FPTAS on graphs of bounded treewidth and a PTAS on planar graphs and more generally on H-minor free graphs. The latter result is shown by adopting a technique of Demaine et al. (2005). We will also show strong NP-hardness of QKP on graphs that are 3-book embeddable, a natural graph class that is related to planar graphs. In addition we will argue that the problem might have a bad approximability behaviour on all graph classes containing large cliques (under certain complexity assumption used for showing hardness results for the densest k-subgraph problem).
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