• A graph consisting of a source \(s\) and sink \(t\) where the edges has non-negative capacity \(c(e)\).

• A flow on an edge is a value \(f(e)\) that does not exceed the value \(c(e)\). \((f(e) \le c(e), \ \forall e \in E\)

• For all vertex \(v \in V\), the sum of incoming flow is eaual to outgoing flow.

• An easier way of viewing network flow is by using water that came out from the source, goes through pipes, and end up in the sink.

• We want to find the maximum amount of flow that can flow through the network
• In the following image, the answer is \(10\)

• Residual graph
• Algorithms:
  • Ford-Fulkerson: A concept, basically check if there exists an Augmenting Path, then flow through it. \(O((V+E)f)\) where \(f\) is max flow.
  • Edmond-Karps: Use BFS to do Ford-Fulkerson \(O(E^2V)\)
  • Dinic: A smarter way with DFS/BFS. \(O(V^2E)\)
  • Dinic is \(O(E\sqrt{V})\) with bipartite graphs.
• Flow algorihms usually run a lot faster than theoretical complexity: \(O(flow)\) as said by taiwanese CPers

• It turns out that for any flow network, we have max flow = min cut.
• We can use this to model even more problems!

Codeforces 1473F - Strange Set

Codeforces - Best Subsequences

Codeforces - Armchair