Alternatives for testing total dual integrality

Total Dual Integrality of Matching Forest Constraints

The b-bibranching Problem: TDI System, Packing, and Discrete Convexity

In this paper, we introduce the b-bibranching problem in digraphs, which is a common generalization of the bibranching and b-branching problems. The bibranching problem, introduced by Schrijver (1982), is a common generalization of the branching and bipartite edge cover problems. Previous results on bibranchings include polynomial algorithms, a linear programming formulation with total dual int...

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:...

Recognizing Totally Dual Integral Systems is Hard

These are notes about Ding, Feng and Zang’s proof [5]. The proof of their result is not new, the only difference with them is the starting point: we work directly on their gadget graph encoding a SAT problem and not on more general graphs. This allows to shortcut some parts of the original proof that become superfluous. After proving their theorem I clarify some points about total dual integral...

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...