Complexity of choosing subsets from color sets Original Research Article

We raise and investigate the algorithmic complexity of the following problem. Given a graph G = (V, E) and p-element sets L(v) for its vertices v ∈ V such that |L(u) ∪ L(v)| ⩾ p + r for all edges uv ∈ E, do there exist q-element subsets C(v) ⊆ L(v) with C(u) ∩ C(v) = 0 for all uv ∈ E? Here p, q, r are positive integers, p ⩾ q and p + r ⩾ 2q. We characterize precisely which triples (p, q, r) admit a polynomial solution, and for which ones the problem is NP-complete. Moreover, it is shown that for some restricted ranges of p and r with respect to q, the existence of subsets C(v) ⊂ L(v) for every collection {L(v)|v ∈ V{, is closely related to Turánʹs problem on uniform hypergraphs.