Lecture22-Notes

Lecture22-Notes - C&O 355 Lecture 22 Notes Nicholas Harvey...

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Nicholas Harvey http://www.math.uwaterloo.ca/~harvey/ 1 Integral Polyhedra We’ve observed the following fact on several occasions. Fact 1.1. Let P R n be a non-empty polyhedron that does not contain a line. For all vectors c such that the LP max ± c T x : x P ² has ﬁnite optimal value, there is an optimal solution that is an extreme point of P . We are interested in the case when the optimal values of this LP are integral. The following theorem was proved by Alan Hoﬀman in 1974. Theorem 1.2. Let P = { x : Ax b } ⊂ R n be a non-empty polyhedron that does not contain a line, and such that the entries of A are integers. Then the following are equivalent. (1): For all vectors c Z n , if the LP max ± c T x : x P ² has ﬁnite optimal value, then the optimal value is an integer. (2): Every extreme point of the polyhedron P is an integral vector. Actually, our hypotheses of this theorem are unnecessarily strong. This theorem also holds if the entries of

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Lecture22-Notes - C&O 355 Lecture 22 Notes Nicholas Harvey...

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