Box-total dual integrality, box-integrality, and equimodular matrices

1 Total Dual Integrality

1 Total Dual Integrality Recall that if A is TUM and b, c are integral vectors, then max{cx : Ax ≤ b} and min{yb : y ≥ 0, yA = c} are attained by integral vectors x and y whenever the optima exist and are finite. This gives rise to a variety of min-max results, for example we derived König’s theorem on bipartite graphs. There are many examples where we have integral polyhedra defined by a syste...

Characterizations of Total Dual Integrality

In this paper we provide new characterizing properties of TDI systems, among others the following: A system of linear inequalities is TDI if and only if there exists a test set for its dual integer linear programs where all positive coordinates are equal to 1, and correspond to a set of linearly independent columns. Cook, Lovász and Schrijver’s result on the recognition – in polynomial time – o...

Total Dual Integrality 1.1 Total Unimodularity

where A and b are rational and the associate dual program min y b s.t. A y = c (2) y ≥ 0 Definition 1 The system of inequalities by Ax ≤ b is Total Dual Integral or TDI if for all integral vectors c the dual program has an integral solution whenever the optimal value is finite. The main result for today is Theorem 1 If Ax ≤ b is TDI and b is integral then P = {x : Ax ≤ b} is integral ∗ ∗ Proof:...

Total Dual Integrality of Matching Forest Constraints

Alternatives for testing total dual integrality

In this paper we provide characterizing properties of TDI systems, among others the following: a system of linear inequalities is TDI if and only if its coefficient vectors form a Hilbert basis, and there exists a test-set for the system’s dual integer programs where all test vectors have positive entries equal to 1. Reformulations of this provide relations between computational algebra and int...