A SOLVABLE CLASS OF QUADRATIC 0-1 PROGRAMMING

We show that the minimum of the pseudo-Boolean quadratic function f(x)=x(T)Qx+c(T)x can be found in linear time when the graph defined by Q is transformable into a combinational circuit of AND, OR, NAND, NOR or NOT logic gates. A novel modeling technique is used to transform the graph defined by Q into a logic circuit. A consistent labeling of the signals in the logic circuit from the set {0, 1} corresponds to the global minimum off and the labeling is determined through logic simulation of the circuit. Our approach establishes a direct and constructive relationship between pseudo-Boolean functions and logic circuits. In the restricted case when all the elements of Q are nonpositive, the minimum of f can be obtained in polynomial time [15]. We show that the problem of finding the minimum of f, even in the special case when all the elements of Q are positive, is NP-complete.