The k-prize-collecting minimum vertex cover problem with submodular penalties (k-PCVCS) is a generalization of the minimum vertex cover problem, which is one of the most important and fundamental problems in graph theory and combinatorial optimization.The k-prize-collecting minimum vertex cover problem with submodular penalties (k-PCVCS) is a generalization of the minimum vertex cover problem, which is one of the most important and fundamental problems in graph theory and combinatorial optimization.[#item_full_content]