CS 411 Fall 2025  >  Outline for November 10, 2025

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CS 411 Fall 2025
Outline
for November 10, 2025

Outline

Iterative Improvement [L Ch 10 Intro]

• What It Is
  • Note: I consider Iterative Improvement in a somewhat more general form than the text.
  • Way of dealing with optimization
    • Finding an optimal solution [considered in the text].
    • Finding a good enough solution.
    • Finding the best solution that resources (e.g., time) allow.
  • Method:
    • Find a feasible solution.
    • Repeat: if possible, improve the feasible solution.
    • Stop when some condition is met.
• When to Stop
  • When no improvements can be found [considered in the text].
  • When a good enough solution has been found.
  • Resource exhaustion (e.g., time limit).
• Challenges
  • Finding a feasible solution to start with
    • Sometimes we can start with a trivial solution (all zeroes?).
  • What improvements do we allow?
  • Distinguishing a local optimum from a global optimum.
    • Does “no improvements found” mean “we have an optimal solution”?

Maximum Flow [L 10.2]

• Introduction
  • Flow network:
    • A digraph.
    • A postitive integer on each arc: the arc’s capacity.
    • There is exactly one source: vertices with no arcs entering.
    • There is exactly one sink: vertices with no arcs leaving.
  • Feasible flow in a flow network:
    • An assignment of a nonnegative number to each arc: the flow on that arc.
    • The flow on each arc is at most the arc’s capacity.
    • Flow conservation requirement: for each vertex other than the source and the sink, the total flow on entering arcs is equal to the total flow on leaving arcs.
  • Value of a feasible flow: the total flow leaving the source (= the total flow entering the sink).
  • Maximum Flow Problem: given a flow network, find a feasible flow with the greatest possible value.
• Augmenting-Path Method
  • Initial flow is zero on every arc.
  • Repeat
    • In the digraph, find a directed path from the source to the sink: an augmenting path. If one cannot be found, then stop.
    • Increase the flow on each arc in this path by the minimum of the capacities of all the arcs.
    • Create a revised flow network showing how the resulting flow can be modified. Reduce capacities, eliminate arcs whose capacity is now zero, and add back arcs as necessary.
• Max-Flow Min-Cut Theorem
  • When the Augmenting-Path Method is finished, the vertices reachable from the source vs. those that are not, form a cut. The total capacity of the arcs across this cut, from the source side to the sink side, equals the value of the flow that has been constructed.
  • Theorem (Max-Flow Min-Cut Theorem). The maximum value of a feasible flow is equal to the minimum capacity of a cut.
• Finding an Augmenting Path
  • Augmenting paths are easy to find, but some ways of finding them result in the A-P method performing poorly.
  • A good way: BFS beginning at source.
• Analysis
  • It is known that, if augmenting paths are found via BFS, then at most \(\frac{|V|\cdot|E|}{2}\) augmentations are required, if capacities are all integers.
  • Let basic ops be single-item operations. A BFS using adjacency lists is \(\Theta(|V|+|E|) = \Theta(|E|)\) for a connected graph.
  • Result: A-P method is \(\Theta(|V|\cdot|E|^2)\).