# Homework Assignment #20.5

## Exercises

### Question 1

7.10 in the textbook

Pages 203 and 204 in the printed textbook will be useful in understanding what is required for finding the smallest (i.e., minimum capacity) $(s-t)$-cut that matches the maximum flow.

### Question 2

Linear Programming and the Maximum Flow Problem: Consider the directed graph $G=(E,V)$. Without loss of generality, let $s$ be the source vertex and $t$ be the sink vertex in $V$. For each edge $(u,v)\in E$, let $c_{uv}$ denote the capacity of that edge. Now, formulate the general maximum flow problem (not a specific instance) as a linear programming problem as follows:

• (a) First, define the variables. (I recommend representing the flows through each edge as your variables.)
• (b) Second, use those variables to formulate all of the necessary elements (i.e., objective function and constraints) of a linear program in algebraic terms.