1 Total Dual Integrality
نویسنده
چکیده
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 system Ax ≤ b but A is not TUM; the polyhedron is integral only for some specific b. We may still ask for the following. Given any c, consider the maximization problem max{cx : Ax ≤ b}; is it the case that the dual minimization problem min{yb : y ≥ 0, yA = c} has an integral optimal solution (whenever a finite optimum exists)? This motivates the following definition: Definition 1 A rational system of inequalities Ax ≤ b is totally dual integral (TDI) if, for all integral c, min{yb : y ≥ 0, yA = c} is attained by an integral vector y∗ whenever the optimum exists and is finite. Remark 2 If A is TUM, Ax ≤ b is TDI for all b.
منابع مشابه
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:...
متن کاملPetersen's Theorem
In this lecture we will cover: 1. Topics related to Edmonds-Gallai decompositions ([Sch03], Chapter 24). 2. Factor critical-graphs and ear-decompositions ([Sch03], Chapter 24). Topics mentioned but covered during subsequent lectures are: 1. The matching polytope ([Sch03], Chapter 25). 2. Total Dual Integrality (TDI) and the Cunningham-Marsh formula ([Sch03], Chapter 25). A detailed reference on...
متن کاملLecture 4 — Total Unimodularity and Total Dual Integrality
Definition 1.1. A matrix A ∈ Zm×n is totally unimodular if the determinant of every square submatrix B ∈ Zk×k equals either −1, 0, or +1. Alternatively, by Cramer’s rule, A ∈ Zm×n is totally unimodular if every nonsingular submatrix B ∈ Zk×k has an integral inverse B−1 ∈ Zk×k. Recall: B−1 = 1 det B B ∗, where B∗ is the adjugate matrix (transpose of the matrix of cofactors) of B. One important c...
متن کاملRecognizing 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...
متن کامل