 By Rainer Burkard, Mauro Dell'Amico, Silvano Martello

ISBN-10: 0898716632

ISBN-13: 9780898716634

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Sample text

The Marriage Theorem and the Existence of Perfect Matchings 19 the two directions: δ + (X) = {(i, j ) ∈ A : i ∈ X, j ∈ X}, δ − (X) = {(i, j ) ∈ A : i ∈ X, j ∈ X}. 7) q(i, j ). (i,j )∈δ + (X) Note that every directed path from the source to the sink contains at least one arc of δ + (X) and that the flow conservation constraints imply f (i, j ) − z(f ) = (i,j )∈δ + (X) f (i, j ). 10. The value z(f ) of an arbitrary (s,t)-flow is always bounded by the value v(C) of an arbitrary (s,t)-cut. Proof. All paths going from s to t use at least one of the arcs of δ + (X).

Theoretical Foundations polyhedra this conjecture is still open. For a survey on the Hirsch conjecture see Klee and Kleinschmidt . A polytope is called Hamiltonian if there exists a path along the edges of the polytope which visits all vertices exactly once and returns to the original starting point. Balinski and Russakoff  show by an explicit construction of such a Hamiltonian cycle that the assignment polytope is Hamiltonian. In a series of papers, Brualdi and Gibson [116, 117, 118, 119, 120] derived further results on the assignment polytope.

Further, the sum of all elements in R is nα. We get n n n nα = i=1 j =1 n n−k rij ≥ n rij + i=1 j =1 rij = (n − k)α + (k + 1)α = (n + 1)α. i=n−k j =1 This implies α = 0 and therefore R = 0. Proof of Birkhoff’s theorem. Let X be an arbitrary doubly stochastic matrix. 19, since otherwise X = 0 contradicting that matrix X is doubly stochastic. 4, there exists a permutation matrix P1 = (pij ) with the following property: if pij = 1, then xij > 0. Let λ1 be the smallest positive entry of X for which pij = 1 holds.