Bicolorings and Equitable Bicolorings of Matrices

Two classical theorems of Ghouila-Houri and Berge characterize total unimodularity and balancedness in terms of equitable bicolorings and bicolorings, respectively. In this paper, we prove a bicoloring result that provides a common generalization of these two theorems. A 0; 1 matrix is balanced if it does not contain a square submatrix with exactly two nonzero entries per row and per column such that the sum of all the entries is congruent to 2 modulo 4. This notion was introduced by Berge [1] for 0; 1 matrices and generalized by Truemper [15] to 0; 1 matrices. A 0; 1 matrix is bicolorable if its columns can be partitioned into blue columns and red columns so that every row with at least two nonzero entries contains either two nonzero entries of opposite sign in columns of the same color or two nonzero entries of the same sign in columns of di erent colors. Berge [1] showed that a 0; 1 matrix A is balanced if and only if every submatrix of A is bicolorable. Conforti and Cornu ejols [6] extended this result to 0; 1 matrices. Cameron and Edmonds [3] gave a simple greedy algorithm to nd a bicoloring of a balanced matrix. In fact, given any 0; 1 matrix A, their algorithm nds either a bicoloring of A or a square submatrix of A with exactly two nonzero entries per row and per column such that the sum of all the entries is congruent to 2 modulo 4. Does this algorithm provide an easy test for balancedness? The answer is no, because the algorithm may nd a bicoloring of A even when A is not balanced. A real matrix is totally unimodular (t.u.) if every nonsingular square submatrix has determinant 1 (note that every t.u. matrix must be a 0; 1 matrix). A 0; 1 matrix A has an equitable bicoloring if its columns can be partitioned into red Dipartimento di Matematica Pura ed Applicata, Universit a di Padova, Via Belzoni 7, 35131, Padova, Italy Graduate School of Industrial Administration, Carnegie Mellon University, Schenley Park, Pittsburgh, Pennsylvania 15213-3890 This work was supported in part by NSF grant DMI-0098427 and ONR grant N00014-97-1-0196.