Hypercylindrically Deduced Cuts in Zero-One Integer Programs

A zero-one mixed integer program in which the integer variables always sum to a constant k in any feasible solution has its solution set contained in the hypercylinder ∑ x i 2 ≦ k . Moreover, integer solutions always satisfy ∑ x i 2 = k , while ∑ x i 2 k for fractional solutions. This situation permits the construction of valid cuts by extending the edges incident at a non-integer extreme point until they intersect the boundary of the hypercylinder; the cut hyperplane is determined by the points where the edge extensions intersect the boundary of the hypercylinder. Generalizations of this special model, due to Glover and Balas, are discussed, as is the use of these cuts in finite algorithms.