Minimization of Half-Products

Since Å xi for binary variables, aiiÅ 0 could be assumed without any loss of generality. 2 x i It is well known that finding the minimum of f is an NP-hard problem (Garey and Johnson 1979); it contains, as special cases, the vertex cover, maximum cut, signed-graph balancing and MAX-2SAT problems. Moreover, for the general minimization problem there exists a constant c ú 0 such that the existence of a (1 0 c)-approximation algorithm would imply P Å NP ; see Arora et al. (1992). This result amplifies the importance of special classes of quadratic pseudo-Boolean functions, for which the minimization problem can be either solved, or at least approximated efficiently. In this paper we introduce a special class, for which the problem of minimization over the set of binary vectors is still difficult, but for which there exists a polynomial time approximation algorithm. The interested reader can find more about related problems, polynomially solvable special cases, algorithms and approximations, e.g., in Boros and Hammer (1991), Deza and Laurent (1992), Padberg (1989) and Palubeckis (1990). Let p Å (p1 , . . . , pn) , q Å (q1 , . . . , qn) ¢ 0 and r Å (r1 , . . . , rn) ¢ 0 be integer vectors, and let us consider the function