18.1 Theorem.
extreme value theorem. A continuous function attains some type of minimum and maximum over an compact set in its domain.
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- Apr 2023
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- Mar 2023
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(2.3)
This is the definition of a reduced costs with potential on the nodes.
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We have seen earlier thatsome feasible circulation Xl in G(XO) satisfies the property that x = XO + Xl.
Applying the flow decomposition theorem from before, we decompose any flows into a sum of cycles flow and paths flows.
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Theorem 3.5 (Flow Decomposition Theorem).
We implicitly assume that the flow is positive due to how the algorithm works.
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Property 3.6. A circulation x can be represented as cycle flow along at mostm directed cycles.
Circulation property. Why is it not along a \(m\) many closed walks?
This is only true from the view of the flow decomposition algorithm and not the LP.
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Potential Functions and Amortized Complexity
Potential function and its application in Amortized Complexity.
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An algorithm is said to be !l(f(n» if for some numbers e / and no and all n ~no, the algorithm takes at least e'f(n) time on some problem instance.
Big Omega definition
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An algorithm is said to be e(f(n» if the algorithm is both O(f(n» and !l(f(n».
Definition of Big Theta.
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An algorithm is said to run in O{f(n» time if for some numbers c and no, thetime taken by the algorithm is at most cf{n) for all n 2: no.
Big O definition
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- Feb 2023
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onvex functions have directional derivatives in alldirections at points in the interior of their domains.
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Theorem 3.7 (Augmenting Cycle Theorem).
Given 2 feasible flow, \(x, x^\circ\), it's possible to construct \(x\) with at most m directed cycles on the residual graph \(G(x^\circ)\).
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3.51
This exercise has been rerefenced due to it being heavily references throughout the text.
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Theorem 3.8 (Negative Cycle Optimality Theorem).
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- Jan 2016
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impedagogy.com impedagogy.com
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In other words, genres of participation are not value-neutral when it comes to issues of equity and opportunity.
power relationships.
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