A MAX-CUT formulation of 0/1 programs

We consider the linear or quadratic 0/1 program P : f∗ = min{cT x + xT Fx : A x = b; x ∈ {0, 1}n }, for some vectors c ∈ Rn , b ∈ Zm , 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 AT A, 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}n is replaced with [0, 1]n (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...

• Type: Journal
• Year: 2016
• Where: ORL
• Authors: Jean B. Lasserre
• Keywords: Operations Research, ORL 2016