Prelude

8.0 - Maximum Flow

• We are going to discuss graph-related algorithms this week:
  • Maximum Flow (Ford Fulkerson Method)
  • Single-Source Shortest Path (Bellman-Ford Algorithm)
  • All-Pairs Shortest Path (Matrix Multiplication Method and Floyd-Warshall Algorithm)

Notation and Terminology

8.0 - Maximum Flow

8.0 - Maximum Flow

• A subgraph of $G = \langle V,E\rangle$ is another graph $G^\prime = \langle V^\prime,E^\prime\rangle$ formed from a subset of the vertices and edges of $G$ ​i.e $E^\prime \subseteq E$ and $V^\prime \subseteq V$
• We also use the term component to refer to a (maximal) connected subgraph 
  • weakly connected vs strongly connected

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Notation and Terminology

Maximum Flow Problem

8.0 - Maximum Flow

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• Suppose that you want to move materials from one location to another location
  • each vertex represents a location
  • each edge represents the connections between two locations

Maximum Flow Problem

8.0 - Maximum Flow

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• The source vertex produces material and the sink vertex consumes them
  • every other vertex, can only propagate the material (so if they receive $n$, then they have to send out $n$ as well)
• The weight of each edge signifies the amount of material that can flow through it

Maximum Flow Problem

8.0 - Maximum Flow

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• The source vertex produces material and the sink vertex consumes them
  • every other vertex, can only propagate the material (so if they receive $n$, then they have to send out $n$ as well)
• The weight of each edge signifies the amount of material that can flow through it

Maximum Flow Problem

8.0 - Maximum Flow

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• In maximum-flow problem, we want to compute the greatest rate at which we can move materials from the source vertex to the sink vertex
• It's not that straightforward: in the above example, the flow out of the source is 29, but you cannot propagate this much material in the network

Maximum Flow Problem

8.0 - Maximum Flow

• It is a relatively common (and complex) problem, but we are going to study a simplified version 
• Examples of applications:
  • goods distribution
  • road networks (traffic control), rail networks
  • water distribution, water pipes
  • airline scheduling
• It is also well-studied:
  • Ford-Fulkerson Algorithm
  • Edmonds-Karp Algorithm
  • Dinic's Algorithm
  • KRT Algorithm

Example

8.0 - Maximum Flow

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Example

8.0 - Maximum Flow

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Here is another flow:

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Definitions

8.0 - Maximum Flow

• A flow network $G = \langle V,E \rangle$  is a directed graph where:
  • each edge $(u,v) \in E$ has a non-negative capacity $c(u,v)$
  • if $(u,v) \in E$, then $(v,u)\not\in E$, i.e. if you have an edge going in one direction, then you cannot have another edge in the reverse direction
  • there is a source vertex $s$ and a sink vertex $t$
• Assumptions (to make our life easier):
  • the graph is connected and each node lies in the path between the source and the sink vertices
  • $|E| \ge |V|-1$ because each vertex has degree at least 1

Definitions

8.0 - Maximum Flow

• A flow in $G$ is a function$f(u,v) : V \times V \rightarrow \mathbb R$ for $u,v \in V$ that satisfies the following two properties:
  • capacity constraint: for all $u,v \in V$,
    • $0\le f(u,v) \le c(u,v)$, i.e. the flow through an edge cannot exceed the capacity of that edge
  • flow conservation: for all $u \in V \setminus \{s,t\}$
    
    ​
    • i.e. except for the source and sink vertices, all inflows to a vertex must equal the outflows from that vertex

$$\sum_{v\in V\setminus \{s,t\}}f(v,u) = \sum_{v\in V\setminus\{s,t\}} f(u,v)$$

Definitions

8.0 - Maximum Flow

this is a flow

this is a different flow

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Definitions

8.0 - Maximum Flow

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$f(s,a) = 11$

$f(s,c) = 8$

$f(a,b) = 11$

$f(b,c) = 3$

$f(c,d) = 11$

$f(d,b) = 7$

$f(b,t) = 15$

$f(d,t) = 4$

the value of this flow is $19$ 

A flow in $G$ is a function $f(u,v) : V \times V \rightarrow \mathbb R$ for $u,v \in V$

 $f(c,a) = 0$ 

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Definitions

8.0 - Maximum Flow

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A flow in $G$ is a function $f(u,v) : V \times V \rightarrow \mathbb R$ for $u,v \in V$

$f(s,a) = 1$

$f(s,c) = 1$

$f(a,b) = 1$

$f(b,c) = 0$

$f(c,d) = 1$

$f(d,b) = 0$

$f(b,t) = 1$

$f(d,t) = 1$

the value of this flow is $2$

$f(c,a) = 0$

Definitions

8.0 - Maximum Flow

• A flow network has many flows, and each flow has its own value $|f|$, defined by:
  • $$|f| = \sum_{v \in V} f(s,v)$$
  • i.e the total flow out of the source vertex (how much can we send out)
  • our goal is to find a flow of maximum value, the maximum flow
• It might be worthwhile for you to stop now and think on how to do this using brute force or also why greedy algorithm won't work

Ford-Fulkerson Method

8.0 - Maximum Flow

• Method:
  • start by setting $f(u,v)$ to $0$ for all $u,v \in V$
  • while there is an augmenting path $p$ in the residual network $G_f$:
    • augment flow $f$ along $p$
• It is not an algorithm, just a method, because it can be implemented in several ways (Edmond-Karp Algorithm is one implementation)
• Let's begin with a run of the method, then we'll go back to these definitions

Ford-Fulkerson Method

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Ingredient #1:

Ford-Fulkerson Method

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Ingredient #2:

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Ingredient #2:

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Ingredient #2:

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Ford-Fulkerson Method

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Residual Network

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$$c_f(u,v)=\begin{cases} c(u,v)-f(u,v)& \text{if $(u,v)\in E$,} \\ f(v,u) & \text{if $(v,u)\in E$,}\\ 0 & \text{otherwise} \end{cases}$$

Residual Network

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$$c_f(u,v)=\begin{cases} c(u,v)-f(u,v)& \text{if $(u,v)\in E$,} \\ f(v,u) & \text{if $(v,u)\in E$,}\\ 0 & \text{otherwise} \end{cases}$$

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$$c_f(u,v)=\begin{cases} c(u,v)-f(u,v)& \text{if $(u,v)\in E$,} \\ f(v,u) & \text{if $(v,u)\in E$,}\\ 0 & \text{otherwise} \end{cases}$$

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$$c_f(u,v)=\begin{cases} c(u,v)-f(u,v)& \text{if $(u,v)\in E$,} \\ f(v,u) & \text{if $(v,u)\in E$,}\\ 0 & \text{otherwise} \end{cases}$$

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Residual Network

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$$c_f(u,v)=\begin{cases} c(u,v)-f(u,v)& \text{if $(u,v)\in E$,} \\ f(v,u) & \text{if $(v,u)\in E$,}\\ 0 & \text{otherwise} \end{cases}$$

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$$c_f(u,v)=\begin{cases} c(u,v)-f(u,v)& \text{if $(u,v)\in E$,} \\ f(v,u) & \text{if $(v,u)\in E$,}\\ 0 & \text{otherwise} \end{cases}$$

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Residual Network

8.0 - Maximum Flow

$$c_f(u,v)=\begin{cases} c(u,v)-f(u,v)& \text{if $(u,v)\in E$,} \\ f(v,u) & \text{if $(v,u)\in E$,}\\ 0 & \text{otherwise} \end{cases}$$

Augmenting Path

8.0 - Maximum Flow

• The residual network tells us how to add flow to the original flow network 
• The idea is to find an augmenting path in the residual network $G_f$ and use this augment the flow $f$ in $G$
  • An augmenting path $p$ is a simple path from $s$ (source) to $t$ (sink) in the residual network $G_f$

8.0 - Maximum Flow

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• The residual network tells us how to add flow to the original flow network 
• The idea is to find an augmenting path in the residual network $G_f$ and use this augment the flow $f$ in $G$
  • remember, don't think of a flow as just one path, a flow is a function that tells us how much capacity we are using on each edge 

Augmenting Path

8.0 - Maximum Flow

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$f(s,a) = 1$

$f(s,c) = 1$

 $f(a,t) = 2$ 

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 $f(c,a) = 1$ 

 $f(c,t) = 0$ 

• The residual network tells us how to add flow to the original flow network 
• The idea is to find an augmenting path in the residual network $G_f$ and use this augment the flow $f$ in $G$
  • remember, don't think of a flow as just one path, a flow is a function that tells us how much capacity we are using on each edge 

Augmenting Path

8.0 - Maximum Flow

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• The residual network tells us how to add flow to the original flow network 
• The idea is to find an augmenting path in the residual network $G_f$ and use this augment the flow $f$ in $G$
  • remember, don't think of a flow as just one path, a flow is a function that tells us how much capacity we are using on each edge