A MAX-CUT formulation of 0/1 programs

We consider the linear or quadratic 0/1 program P : f = min{cx + xFx : Ax = b; x ∈ {0, 1}}, for some vectors c ∈ R, b ∈ Z, some matrix A ∈ Zm×n and some real symmetric matrix F ∈ Rn×n. We show that P can be formulated as a MAX-CUT problem whose quadratic form criterion is explicit from the data of P. In particular, to P one may associate a graph whose connectivity is related to the connectivity of the matrix F and AA, and P reduces to finding a maximum (weighted) cut in such a graph. Hence the whole arsenal of approximation techniques for MAX-CUT can be applied. On a sample of 0/1 knapsack problems, we compare the lower bound on f∗ of the associated standard (Shor) SDP-relaxation with the standard linear relaxation where {0, 1} is replaced with [0, 1] (resulting in an LP when F = 0 and a quadratic program when F is positive definite). We also compare our lower bound with that of the first SDP-relaxation associated with the copositive formulation of P.