Fair allocation of indivisible goods under conflict constraints

We consider the problem of fairly allocating indivisible items to several agents, each of them having different profit valuations of items. An additional condition allows an incompatibility relation between pairs of items represented by a conflict graph. Hence, every feasible allocation of items to the agents corresponds to a partial coloring, that is, a collection of pairwise disjoint independent sets. The sum of profits of vertices/items assigned to one color/agent should be optimized in a maxi-min sense. The problem is studied for different classes of conflict graphs. On the one hand we show strong NP-hardness for bipartite graphs and their line graphs, on the other hand we derive pseudo-polynomial solution algorithms for cocomparability graphs and biconvex bipartite graphs.