## Integer rounding and combinatorial max-min theorems |

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### Contents

THE INTEGRALITY PROPERTY FOR PACKING POSITIVE PARTS | 22 |

THE IRD PROPERTY FOR THE VERTEXPACKING PROBLEM | 40 |

SOME OPTIMIZATION PROBLEMS ON BRANCHINGS | 57 |

### Common terms and phrases

a-critical graphs algebraic proof Ax _ Baum and Trotter bipartite bipartite graph blocking matrix blocking pair branchings in G cocycle combinatorial max-min theorems common independent sets complete graph components convex hull covering the edges defined denoted directed graph edge directed edge set edges of G Edmonds extreme points forward edges Fulkerson G contains graph G incidence vectors induced subgraph inessential integer programming problems integer rounding holds integrality property IRD holds IRD property Lemma Let G Let the rows linear programming problem LP relaxation m x n matroids max{l-y min-max theorem nonnegative integral vector odd cycles odd subdivision optimal solution packing and covering packing problem pair of matrices polymatroid polynomial time algorithm property holds proved r-branchings Robacker's theorem row space rows of matrix set of edges simple paths standard representative matrix subgraph of G subset Theorem III.2.2 totally unimodular matrix vertex vertex-packing problem vertices x e Rn