Integer Rounding and Polyhedral Decomposition for Totally Unimodular Systems

Abstract

It is shown that total unimodularity of a matrix A is equivalent to the following decomposition property for the polyhedron P(b) = {x ∈ R ⁿ₊ : Ax ≦ b}: for each integral vector b and each integral scalar k > O, every integral vector of P(kb) is the sum of k integral vectors of P(b). For a given totally unimodular matrix A and integer vector b let the minimal integral vectors of the equality system Q(b)={x∈R ⁿ₊ :Ax=b} be given by the rows of matrix B and (when Q(b) is bounded) let the maximal integral vectors of Q(b) be given by the rows of matrix C. It is shown that Q(b) satisfies the above decomposition property if and only if, for every integral vector w ≥ 0, the objective value of a best integral solution to the combinatorial packing problem max{1·y:yB ≤ w, y ≥ O} is simply the round-down to the nearest integer of the optimal solution value. Similarly, when Q(b) is bounded, decomposition for Q(b) is equivalent to the requirement that, for every integral vector w ≥ 0, the objective value of a best integral solution to the combinatorial covering problem min{1·y:yC ≥ w, y ≥ o} is given by rounding the optimal solution value up to the nearest integer.

Research partially supported by N.S.F. Grant ENG 76-09936

Research partially supported by N.S.F. Grant ENG 76-09936 and by Sonderforschungsbereich 21 (DFG), Institut für Operations Research, Universität Bonn, Bonn