The multiple knapsack problem with compatible bipartite graphs

The multiple knapsack problem is to pack some items into given knapsacks, such that the sum of the knapsack profits is maximized. This paper is concerned with a variant of the multiple knapsack problem, called the multiple knapsack problem with compatible bipartite graph (MKPCBG), where two items can be packed into the same knapsack only if their corresponding vertices are adjacent in the given compatible bipartite graph. Under two different objectives, we prove that the MKPCBG problem is strongly NP-hard, design some 1/2-approximation algorithms, and design two optimal algorithms for the special case where all knapsacks have the same capacity.