Partitioning a matrix with non-guillotine cuts to minimize the maximum cost

We consider the problem of partitioning a matrix of m rows and n columns of non-negative integers into M smaller submatrices. With each submatrix is associated a cost equal to the sum of its elements. The objective is to minimize the cost of the submatrix of maximum cost. We present a (O-l)-integer programming formulation of the problem and three different lower bounds. A heuristic procedure for finding a valid upper bound to the optimal solution cost is also described. Problem reduction tests derived from both the original problem and the lower bounds are given. Lower bounds and reduction tests are used in a tree search algorithm in order to find the optimal solution to the problem. Computational results on a number of randomly generated test problems are presented. © 2002 Elsevier Science B.V. AU rights reserved.