Max-flow via Linear Programming
Variables for every . Objective function: . Constraints: for every , , for every , .
Standard form for LP
variables , constraints , objectives , and . We want to maximize subject to . (for example refer to the written notes)
Primal LP Dual LP
For a LP, optimum is achieved at a vertex of the feasible region except if
- it’s infeasible (e.g.: & ), so feasible region is empty
- it’s unbounded (e.g.: s.t. )
Weak duality theorem
For a LP whose primal and dual are feasible, their optimum and satisfy
Consequence: if we have a feasible of primal LP and a feasible of dual LP, and , then and are optimal solutions.
Strong duality theorem
If primal LP has optimal and dual LP has optimal , then .
If primal LP is unbounded, then dual LP is infeasible.
If dual LP is unbounded, then primal LP is infeasible.
Given points , find the line which minimizes the maximum distance.
Variables: . Goal: minimize where for all ,
LP Problem: variables , minimize s.t.
In standard form, maximize , s.t.
How to get nonnegative constraints? Make variables , and replace .
New LP becomes: maximize s.t.