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 (G) equals its clique partition number (G) even when some minimum clique partition of G is given. This implies that any (G)-upper bound provably better than (G) is NPhard 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 (G) 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 (G) h always holds. Thus, QCP is satisfiable if and only if (G) = (G) = h. Computing the Lovász number θ(G) we can detect QCP unsatisfiability at least when (G) 0 gap recognition, with one minimum clique partition of G known.