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usaco-guide/content/6_Plat/Flows.mdx
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---
id: max-flow
title: "Maximum Flow"
author: Benjamin Qi
prerequisites:
- Gold - Breadth First Search
description: Introduces maximum flow as well as flow with lower bounds.
frequency: 1
---
Has not been the solution to any platinum problem, but appeared in old gold.
## Maximum Flow
- [CSES Download Speed](https://cses.fi/problemset/task/1694)
### Tutorial
- [CPC.10](https://github.com/SuprDewd/T-414-AFLV/tree/master/10_graphs_3_network_flow)
- CPH (20, Flows & Cuts)
- Topics
- Bipartite Matching?
- Edmonds-Karp Algorithm
- Min-Cut Max-Flow Theorem
- [Closure Problem](https://en.wikipedia.org/wiki/Closure_problem)
### Dinic's Algorithm
* Hopcroft-Karp Bipartite Matching?
<optional-content title="Faster Flow">
There exist faster flow algorithms such as **Push-Relabel**. Also see the following blog post:
- [Chilli: Dinic's with Scaling](https://codeforces.com/blog/entry/66006)
However, the standard implementation of Dinic's is (almost) always fast enough.
</optional-content>
### Problems
- CSES
- [School Dance](https://cses.fi/problemset/task/1696)
- [Coin Grid](https://cses.fi/problemset/task/1709)
- [Distinct Routes](https://cses.fi/problemset/task/1711)
- Other
- [Goods Transportation](http://codeforces.com/problemset/problem/724/E) [](52)
- [ARC 85 MUL](http://arc085.contest.atcoder.jp/tasks/arc085_c) [](67)
- [Fashion](https://csacademy.com/contest/rmi-2017-day-1/task/fashion/statement/) [](95)
- [Fast Flow](http://www.spoj.com/problems/FASTFLOW/) [](107)
- [Card Game](http://codeforces.com/problemset/problem/808/F) [](135)
## Flow with Lower Bounds
### Tutorial
- [CMU Slides](https://www.cs.cmu.edu/~ckingsf/bioinfo-lectures/flowext.pdf)
### Problems
- [Diverse Singing](https://codeforces.com/group/ZFgXbZSjvp/contest/274398/problem/C)
- [Captain America](https://codeforces.com/contest/704/problem/D)
- [Incorrect Flow](https://codeforces.com/contest/708/problem/D)
- [Multi-Path Story](https://atcoder.jp/contests/jag2013summer-day4/tasks/icpc2013summer_day4_i)
- [WAC 4 - Hungry Squirrels](https://dmoj.ca/problem/wac4p6)