Totally-Balanced and Greedy Matrices

Totally-balanced and greedy matrices are $( 0,1 )$-matrices defined by excluding certain submatrices. For a $n \times m \,( 0,1)$-matrix A we show that the linear programming problem $\max \{ by \mid yA\leqq c,0\leqq y\leqq d \}$ can be solved by a greedy algorithm for all $c\geqq 0$, $d\geqq 0$ and $b_1 \geqq b_2 \geqq \cdots \geqq b_n \geqq 0$, if and only if A is a greedy matrix. Furthermore we show constructively that if b is an integer, then the corresponding primal problem $\min \{ cx + dz \mid Ax + z\geqq b,x\geqq 0,z\geqq 0 \}$ has an integer optimal solution. A polynomial-time algorithm is presented to transform a totally-balanced matrix into a greedy matrix as well as to recognize a totallybalanced matrix. This transformation algorithm together with the result on greedy matrices enables us to solve a class of integer programming problems defined on totally-balanced matrices. Two examples arising in tree location theory are presented.