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  1. Apr 2023
    1. 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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  2. Mar 2023
    1. (2.3)

      This is the definition of a reduced costs with potential on the nodes.

    2. 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.

    3. Theorem 3.5 (Flow Decomposition Theorem).

      We implicitly assume that the flow is positive due to how the algorithm works.

    4. 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.

    5. Potential Functions and Amortized Complexity

      Potential function and its application in Amortized Complexity.

    6. 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

    7. 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.

    8. 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

  3. Feb 2023
    1. onvex functions have directional derivatives in alldirections at points in the interior of their domains.

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    1. 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 \(G(x^\circ)\).

    2. 3.51

      This exercise has been rerefenced due to it being heavily references throughout the text.

    3. Theorem 3.8 (Negative Cycle Optimality Theorem).
  4. Jan 2016