Cutting Planes in Combinatorics
European journal of combinatorics (Print)
In chapter 26 of his book, George Dantzig presented side by side (i) a number of difficult mathematical problems reducible to integer linear programming problems, and (ii) Gomory's cutting-plane method for solving integer linear programming problems. In the same spirit, we illustrate the use of cutting-plane arguments in solutions of combinatorial problems. To GEORGE DANTZIG FOR HIS 70th BIRTHDAY In Chapter 26 of his book , George Dantzig presented side by side (i) a number of difficult
... matical problems reducible to integer linear programming problems, and (ii) Gornory's cutting-plane method for solving integer linear programming problems. In the same spirit, we are going to illustrate the use of cutting-plane arguments in the solution of combinatorial problems. To approach the subject gently, let us first consider the following problem from recreational mathematics : How many diamonds can be packed in a Chinese checkerboard? This board consists of two order 13 triangular arrays of holes, overlapping in an order 5 hexagon, 121 holes in all. A diamond consists of four marbles that fill four adjacent holes. Fitting 27 diamonds onto the board is quite easy (see Figure 1) ; showing that 28 diamonds will not fit may get a little more complicated. However, as soon as the problem is stated in integer linear programming terms, an elegant argument to establish the upper bound suggests itself. The integer linear programming formulation is straightforward: number the holes as 1,2, ... ,121, think of each diamond D as a set of four holes, and describe each packing of diamonds in the board by setting XD = 1 if D is in the packing, and XD =0 otherwise. In this notation, our problem is to maximize L XD subject to L (XD: i E D)~1, for all i = 1, 2, ... ,121, XD = 0 or 1 for all D. (Feasible solutions of (1) are in a one-to-one correspondence with packings; in this correspondence, the objective function of (1) counts the diamonds in the packing.) As usual, one may begin to solve (1) by solving its 'LP relaxation', the linear programming problem maximize LXD subject to L (XD: i E D)~1, for all i = 1, 2, ... , 121, XD ;;. 0 for all D. Since each feasible solution of (1) is a feasible solution of (2), the optimal value of (2) provides an upper bound on the optimal value of (1). As luck would have it, the optimal value of (2) is 27'5 and so the optimal value of (1), being an integer, cannot exceed 27.