A polynomial time primal network simplex algorithm for

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A polynomial time primal network simplex algorithm for minimum cost flows James B. Orlin

A polynomial time primal network simplex algorithm for minimum cost flows James B. Orlin Presented by Tal Kaminker

Reminder – minimum cost flow •

Reminder – minimum cost flow •

overview •

overview •

Non-degeneracy assumption •

Non-degeneracy assumption •

notations •

notations •

The regular network simplex algorithm •

The regular network simplex algorithm •

Potential - reminder •

Potential - reminder •

rooted in-tree •

rooted in-tree •

premultipliers •

premultipliers •

premultipliers •

premultipliers •

premultipliers •

premultipliers •

premultipliers •

premultipliers •

premultipliers network simplex algorithm •

premultipliers network simplex algorithm •

modify-premultipliers •

modify-premultipliers •

Lemma: The premultiplier algorithm maintains a legal vector of premultipliers at every step. Proof:

Lemma: The premultiplier algorithm maintains a legal vector of premultipliers at every step. Proof: • simplex-pivot maintains a legal tree

Proof cont. •

Proof cont. •

Proof cont. •

Proof cont. •

premultipliers network simplex algorithm Lemma: Each call of modify-premultipliers strictly increases the number of

premultipliers network simplex algorithm Lemma: Each call of modify-premultipliers strictly increases the number of eligible nodes. Theorem: The premultiplier algorithm is a special case of the network simplex algorithm. As such, it solves the minimum cost flow problem in a finite number iterations.

cost-scaling version of premultipliers alg. •

cost-scaling version of premultipliers alg. •

• cost-scaling algorithm

• cost-scaling algorithm

correctness Theorem: The algorithm stops after finite amount of iterations and at the end

correctness Theorem: The algorithm stops after finite amount of iterations and at the end yields the optimal flow Proof: Almost the same as the regular premultiplier algorithm

Differences between the two algorithms •

Differences between the two algorithms •

outline •

outline •

Thank you Questions?

Thank you Questions?

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