Solutions for the knapsack problem with conflict and forcing graphs of bounded clique-width

The 0–1-knapsack problem is a well-known NP-hard problem in combinatorial optimization. We consider the extensions to the knapsack problem with conflict graph (KCG) and the knapsack problem with forcing graph (KFG). KCG has first been introduced by Yamada et al. and represents incompatibilities between items of the knapsack instance. KFG has been introduced by Pferschy and Schauer and represents the necessity of items for other items. Within this paper we provide pseudo-polynomial solutions for KCG and KFG with co-graphs as conflict and forcing graphs and extend these solutions to conflict and forcing graphs of bounded clique-width. Our solutions are based on dynamic programming using the tree-structure representing the conflict graph and the forcing graph. Further we conclude fully polynomial time approximation schemes for KCG on conflict graphs of bounded clique-width and KFG on forcing graphs of bounded clique-width. This generalizes the known results for conflict graphs and forcing graphs of bounded tree-width of Pferschy and Schauer.