Total Dual Integrality 1.1 Total Unimodularity
نویسنده
چکیده
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: We proceed by contradiction. Consider a vertex x of P such that xj ∈ Z. We can ∗ construct an integral c such that x is the optimal solution corresponding to c by picking a rational ∗ c in the optimal cone of x and scaling. Consider ĉ = c + 1 ej where q is an integer. Since the cone q ∗ is full dimensional, ĉ will still be in the optimality cone of x for q sufficiently large. Now it follows ∗ ∗ ∗ ∗ ∗ that qĉ = qc + ej and thus (qĉ) x − (qc) x = xj ∈ Z. This means that either (qĉ) x or (qc) x are not integral which contradicts the assumption of total dual integrality. � Note that the converse doesn’t generally hold. We can have Ax ≤ b integral with b an integral vector, but the system is not TDI.
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