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

Abstract

We show that for a graph G it is NP-hard to decide whether its independence number α ( G ) equals its clique partition number χ ¯ ( G ) even when some minimum clique partition of G is given. This implies that any α ( G ) -upper bound provably better than χ ¯ ( G ) 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 α ( 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 χ ¯ ( G ) ⩽ h always holds. Thus, QCP is satisfiable if and only if α ( G ) = χ ¯ ( G ) = h . Computing the Lovász number ϑ ( G ) we can detect QCP unsatisfiability at least when χ ¯ ( G ) < h . In the other cases QCP reduces to χ ¯ ( G ) - α ( G ) > 0 gap recognition, with one minimum clique partition of G known.