Fabio D’Andreagiovanni

1. Lecture Notes on Total Unimodularity Fabio D’Andreagiovanni

2. Total Unimodularity Definition 1: an (m x n) matrix A is unimodular iff for every (m x m) square submatrix B of A it holds det (B) Î {-1, 0, 1} Definition 2: an (m x n) matrix A is totally unimodular iff for every (p x p) square submatrix B of A with p>0 it holds: det (B) Î {-1, 0, 1} Unimodular but not totally unimodular Unimodular and totally unimodular Not Unimodular

3. On unimodularity and integral vertices Theorem THF1: letAbe an (m x n) integer matrixA such that rank(A) = m. The followingstatementsareequivalent: A is unimodular The verticesofthepolyhedronP = {xÎRn: Ax=b, x³0n} areintegralforeverybÎZm Every (m x m) squaresubmatrixB of A thatis non-singularhas an integer inverse matrix B-1 Proof: we prove the equivalence showing that Þ Þ Þ (3 1) (1 2) (2 3)

4. Þ On unimodularity and integral vertices (1 2) If A is unimodular then the vertices of the polyhedron P = {xÎRn: Ax=b, x³0n} are integral for every bÎZm Proof: xo vertex of P = {xÎRn: Ax=b, x³0n} xobasic feasible solution there exists (m x m) square submatrix B of A with det(b)=0 xBÎRm xNÎRn-m such that x = and A = (B N) xoBÎRm xoNÎRn-m B-1b 0n-m We have Ax=b BxB+NxN=b and xo= = ³ 0n Þ Þ Þ Þ Þ B-1= adj(B) / det(B) where adj(B) is the adjunct matrix of B: ÞB-1 integer matrix A integer matrix Þ Þ B-1bÎZm for every bÎZm A unimodular matrix |det(B)|=1 Þ x0ÎZn for every bÎZm

5. Þ On unimodularity and integral vertices (2 3) If the vertices of the polyhedron P = {xÎRn: Ax=b, x³0n} are integral for every bÎZm , then every (m x m) square submatrix B of A that is non-singular has an integer inverse matrix B-1 Proof: Let B be an (m x m) square submatrix of A with det(b)=0 and denote by B-1i the i-th column of the corresponding inverse matrix B-1. • We prove that the generic i-th column B-1i has integer components: • Let tÎZm be an integer vector such that t+B-1i ³ 0m • Let b(t)=Bt+ei with ei being the i-th unit vector, then B-1b(t) 0n-m B-1(Bt+ei) 0n-m t + B-1ei 0n-m t + B-1i 0n-m ³ 0n = = = basic feasible solution of the system Ax=b(t), x ³ 0n Þ Vertex of P = {xÎRn: Ax=b(t), x ³ 0n} Þ t+B-1i is an integer vector B-1i is an integer vector Þ Þ

6. Þ On unimodularity and integral vertices (3 1) If every (m x m) square submatrix B of A that is non-singular has an integer inverse matrix B-1, thenA is unimodular Proof: If B is an (m x m) square submatrix of A with det(b)=0, then B-1 is an integer matrix |det(B)| and |det(B-1)| are integers Þ Since |det(B)| |det(B-1)| = |det(BB-1)| = 1 |det(B)| =|det(B-1)| = 1 Þ A is unimodular Þ Quod erat demonstrandum

7. Vertices and standard form Theorem THF2: Let A be an (m x n) matrix and b an m-dimensional vector. x0 is a vertex of the polyhedronP = {xÎRn: Ax=b, x³0n} if and only if the point is a vertex of the polyhedron PSTD = {(x,y)ÎRn+m: Ax+Is=b, x³0n, y³0m} x0ÎRn y0ÎRm x0 b-Ax0 =