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 γ
r
:=n
−1max{x
T
VV
T
x: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 γ
r
<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.
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Malik, U., Jaimoukha, I., Halikias, G. et al. On the gap between the quadratic integer programming problem and its semidefinite relaxation. Math. Program. 107, 505–515 (2006). https://doi.org/10.1007/s10107-005-0692-2
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DOI: https://doi.org/10.1007/s10107-005-0692-2