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

Consider the semidefinite relaxation (SDR) of the quadratic integer program (QIP): γ := maxx∈{−1,1}n xT Qx ≤ minD−Qo0 trace(D) =: γ̄ where Q is a given symmetric matrix and D is diagonal. In this paper we consider the relaxation gap γ̄ − γ. We establish the uniqueness of the solution of the semidefinite relaxation problem and prove that γ = γ̄ if and only if γr := 1 n maxx∈{−1,1}n xT V V T x = 1 where V is an orthonormal matrix whose columns span the (r– dimensional) null space of D−Q and where D is the unique solution for the relaxation problem. We also give a test for establishing whether γ = γ̄ that involves 2r−1 function evaluations. In the case that γr < 1 we give a new upper bound on γ which is tighter than γ̄. Thus we show that ‘breaching’ the semidefinite relaxation gap for the quadratic (-1,1) integer programming problem is as difficult as the solution of a quadratic (-1,1) integer program 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.