Lecture 04-21-17

Exam Review

1. Hamilton path -> cycle
   • In NP:
   • Easy to prove NP (just verify uses each once) - Reduction
   • G’ = G + Universal vertex
2. P/NP/NPC concepts
3. K + 1 bins, Edge Double Cover
   • NIFF was 4/3
   • Other part: give a proof where on a certain example it performs k optimal
4. Floyd’s algorithm (all pairs shortest path)

Max Flow, Matching

• Can measure flow across any cut

Ford & Fulkerson’s

• Flow: between 0 and capacity of edge obviously
• G_f: residual flow graph (how we can change the flow on G)
• For any edge: can increase the flow by C-F and decrease by F
• Represented by a back edge (has capacity F, flow F)
• Why would you want to reduce the flow on an edge?
  • Because this algorithm is based on local improvement
  • Argue that when the algorithm stops, we have the maximum flow
  • Cases where needs to be decreased to find the maximum

Ford & Fulkerson’s algorithm

• At any point in time, have a graph G
  • Everything has a capacity and flow
  • Have a graph G’ which will be used to find the maximum
• Start with 0 flow
  • (Works with any legal starting flow)
• While there is a path from S to T in G_f
  • Increase the flow along that path
  • Change by the smallest capacity on that path
  • So far only forward edges:
    • Simply add to the flow whenever forward edge
  • Back edge:
    • Reduce the flow on the edge

G' = graph with 0 flow                  // Works with any legal starting flow
while exists path from S to T:
    cap = min(capacities of e in E)
    for edge (u, v) in p:
        if (u, v) e E:                  // Edges in the original graph
            (u, v).f += cap
        else:
            (v, u).f -= cap

• Argue that when it stops: flow is maximum
  • Take a cut (we can measure across any cut)
• Running time: O(Very very bad)
  • Only if you are an idiot about choosing paths from S to T