In [7]:
# %load ../../preconfig.py
%matplotlib inline
import matplotlib.pyplot as plt
import seaborn as sns
sns.set(color_codes=True)
plt.rcParams['axes.grid'] = False

#import numpy as np
#import pandas as pd

#import sklearn

#import itertools

import logging
logger = logging.getLogger()

def show_image(filename, figsize=None, res_dir=True):
    if figsize:
        plt.figure(figsize=figsize)

    if res_dir:
        filename = './res/{}'.format(filename)

    plt.imshow(plt.imread(filename))

26 Maximum Flow

each directed edge in a flow network like a conduit for the material: Each conduit has a stated capacity, vertices are conduit junctions.

In the maximum-flow problem, we wish to compute the greatest rate at which we can ship material from the source to the sink without violating any capacity constraints.

26.1 Flow networks

Let $G = (V, E)$ be a flow network with a capacity function $c$.

Let $s$ be the source of the network, and let $t$ be the sink.

A flow in $G$ is a real-valued function $f : V \times V \to \mathcal{R}$ that satisfies the following two properties:

  • Capacity constraint: $0 \geq f(u, v) \geq c(u, v) : \forall u, v \in V$

  • Flow conservation: $$\displaystyle \sum_{v \in V} f(v, u) = \sum_{v \in V} f(u, v)$$

  • real-world flow problem may violates our assumption:
    modeling problems with antiparalled edges.

  • networks with multiple sources and sinks:
    add a supersourse and a supersink.

In [3]:
plt.imshow(plt.imread('./res/fig26_1.png'))
Out[3]:
<matplotlib.image.AxesImage at 0x114578eb8>
In [4]:
# Exercise

26.2 The Ford-Fulkerson method

The Ford-Fulkerson method depends on three important ideas: residual networks, augmenting paths, and cuts.

FORD-FULKERSON-METHOD(G, s, t)
initialize flow f to 0

while there exists an augmenting path p in the residual network G_f
    augment flow f along p

return f

Residual networks

the residual network $G_f$ consists of edges with capacitites that represent how we can change the flow on edges of $G$.

we define the residual capacity $c_f(u, v)$ by \begin{equation} 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} \end{equation}

the residual network of $G$ induced by $f$ is $G_f = (V, E_f)$, where $$E_f = \{ (u, v) \in V \times V : c_f(u, v) > 0 \}$$

If $f$ is a flow in $G$ and $f'$ is a flow in the corresponding residual network $G_f$, we define$f \uparrow f'$, the augmentation of flow $f$ by $f'$: \begin{equation} (f \uparrow f')(u, v) = \begin{cases} f(u, v) + f'(u, v) - f'(v, u) \, & \text{ if } (u, v) \in E \\ 0 \, & \text{ otherwise} \end{cases} \end{equation}

In [9]:
show_image('fig26_4.png', figsize=(8,12))

Augmenting paths

an augmenting path $p$ is a simple path from $s$ to $t$ in the residual network $G_f$.

$$c_f(p) = \min \{ c_f(u, v): \text{ $(u, v)$ is on $p$} \}$$

Cuts of flow networks

The Ford-Fulkerson method repeatedly augments the flow along augmenting paths until it has found a maximum flow.

The max-flow min-cut theorem tells us that a flow is maximum if and only if its residual network contains no augmenting path.

The basic Ford-Fulkerson algorithm

In [10]:
show_image('ford.png')
In [11]:
show_image('fig26_6.png')

The Edmonds-Karp algorithm

We can improve the boudn of FORD-FULKERSON by finding the augmenting path $p$ in line 3 with a breadth-first search.

In [13]:
#Exercise

26.3 Maximum bipartite matching

a maximum matching in a bipartite graph $G$ corresponds to a maximum flow in its corresponding flow network $G'$, and we can therefore compute a maximum matching in $G$ by running a maximum-flow algorithm on $G'$.

In [14]:
show_image('fig26_8.png')
In [15]:
#Exercise

26.4 Push-relabel algorithms