On the set covering polytope: I. All the facets with coefficients in {0,1,2}

While the set packing polytope, through its connection with vertex packing, has lent itself to fruitful investigations, little is known about the set covering polytope. The authors characterize the class of valid inequalities for the set covering polytope with coefficients equal to 0, 1 or 2, and give necessary and sufficient conditions for such an inequality to be minimal and to be facet defining. They show that all inequalities in the above class are contained in the elementary closure of the constraint set, and that 2 is the largest value of k such that all valid inequalities for the set covering polytope with integer coefficients no greater than k are contained in the elementary closure. The authors point out a connection between minimal inequalities in the class investigated and certain circulant submatrices of the coefficient matrix. Finally, they discuss conditions for an inequality to cut off a fractional solution to the linear programming relaxation of the set covering problem and to improve the lower bound given by a feasible solution to the dual of the linear programming relaxation.