On the gap between the quadratic integer programming problem and its semidefinite relaxation

Abstract

Consider the semidefinite relaxation (SDR) of the quadratic integer program (QIP): ** where Q is a given symmetric matrix and D is diagonal. We consider the SDR gap **. We establish the uniqueness of the SDR solution and prove that ** if and only if γr:=n−1max{xTVVTx:x ∈ {-1, 1}n}=1 where V is an orthogonal matrix whose columns span the (r–dimensional) null space of D−Q and where D is the unique SDR solution. We also give a test for establishing whether ** that involves 2r−1 function evaluations. In the case that γr<1 we derive an upper bound on γ which is tighter than **. Thus we show that `breaching' the SDR gap for the QIP problem is as difficult as the solution of a QIP with the rank of the cost function matrix equal to the dimension of the null space of D−Q. This reduced rank QIP problem has been recently shown to be solvable in polynomial time for fixed r.