On NP-hardness of the clique partition—Independence number gap recognition and related problems

We show that for a graph G it is NP-hard to decide whether its independence number image equals its clique partition number image even when some minimum clique partition of G is given. This implies that any image-upper bound provably better than image is NP-hard to compute. To establish this result we use a reduction of the quasigroup completion problem (QCP, known to be NP-complete) to the maximum independent set problem. A QCP instance is satisfiable if and only if the independence number image of the graph obtained within the reduction is equal to the number of holes h in the QCP instance. At the same time, the inequality image always holds. Thus, QCP is satisfiable if and only if image. Computing the Lovász number image we can detect QCP unsatisfiability at least when image. In the other cases QCP reduces to image gap recognition, with one minimum clique partition of G known.