By Jon Lee

ISBN-10: 0521010128

ISBN-13: 9780521010122

ISBN-10: 0521811511

ISBN-13: 9780521811514

Jon Lee makes a speciality of key mathematical rules resulting in invaluable versions and algorithms, instead of on info buildings and implementation info, during this introductory graduate-level textual content for college students of operations learn, arithmetic, and laptop technology. the perspective is polyhedral, and Lee additionally makes use of matroids as a unifying proposal. subject matters contain linear and integer programming, polytopes, matroids and matroid optimization, shortest paths, and community flows. difficulties and workouts are integrated all through in addition to references for extra examine.

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**Extra resources for A First Course in Combinatorial Optimization **

**Example text**

If x β ≥ 0 then the basic solution is feasible to P . Depending on whether the basic solution x ∗ is feasible or optimal to P , we may refer to the basis β as being primal feasible or primal optimal. The dual linear program of P is m yi bi min i=1 (D ) subject to: m yi ai j ≥ c j , for i = 1, 2, . . , n. i=1 Associated with the basis β is a potential solution to D . cls 22 December 11, 2003 16:30 Char Count= 0 0 Polytopes and Linear Programming associated with β is the unique solution y1∗ , y2∗ , .

Proof. Let x be a feasible basic solution of P , with corresponding basis β and nonbasis η. Suppose that x = λx 1 + (1 − λ)x 2 , with x 1 , x 2 ∈ P , and 0 < λ < 1. Because xη = 0, xη1 , xη2 ≥ 0, xη = λxη1 + (1 − λ)xη2 , we must have xη1 = xη2 = xη = 0. Then we must have Aβ xβl = b, for l = 1, 2. However, this system has the unique solution xβ . Therefore, xβ1 = xβ2 = xβ . Hence, x 1 = x 2 = x. Conversely, suppose that x is an extreme point of P . Let φ := j ∈ {1, 2, . . , n} : x j > 0 . We claim that the columns of Aφ are linearly independent.

Cls 42 T1: IML December 11, 2003 16:30 Char Count= 0 0 Polytopes and Linear Programming where Aiβ = Aβ1 , . . , Aβi−1 , b, Aβi+1 , . . , Aβm . Because Aiβ is integer valued, we have that det(Aiβ ) is an integer. Also, because A is totally unimodular and Aβ is nonsingular, we have det(Aβ ) = ±1. Therefore, xβ∗ is integer valued. A “near-converse” result is also easy to establish. Theorem (Integrality implies unimodularity). Let A be an integer matrix. If the extreme points of {x ∈ Rn : Ax ≤ b, x ≥ 0} are integer valued, for all integer vectors b, then A is totally unimodular.

### A First Course in Combinatorial Optimization by Jon Lee

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