1. Lecture 19 Linear Program

2. Recap Shortest path Minimum spanning tree Maximum matching
Common: Optimizing some objective (length of paths, weight of the tree, number of matches)
Very different techniques
Hope: A common technique that can solve all problems

3. Linear Relationships Inequalities that are linear in all parameters.
Example. If d[u] = shortest path distance from s to u, then for any edge (u,v)
𝑑 𝑣 ≤𝑑 𝑢 +𝑤(𝑢,𝑣)

4. Linear Relationships
Example 2: Let xi,j = 1 if course i is matched to classroom j, and 0 otherwise.
Each classroom is matched to at most one course.
∀𝑗 𝑖=1 𝑛 𝑥 𝑖,𝑗 ≤1

5. Linear Program
Optimize a linear function (objective), under a set of linear inequality constraints.
max 2𝑥+𝑦 𝑥≥0 𝑦≥0 𝑥+𝑦≤1

6. Geometric Interpretation
Linear inequality  Half planes x y x y x y
𝑥+𝑦≤1
𝑥≥0
𝑦≥0

7. Geometric Interpretation
System of linear inequalities  intersections
Green: Feasible set. Point in Green: feasible solution. x y

8. Geometric Interpretation
Objective function  Direction of gravity x y x y
max 2𝑥+𝑦

9. Geometric Interpretation
Optimal Point  Lowest point
Optimal solution: (x, y) = (1, 0), value = 2. x y

10. Canonical Form
min 𝑐,𝑥 𝐴𝑥≥𝑏 𝑥≥0 c x A x b ≥

11. Converting to Canonical Form
Equality constraints (e.g. x+y = 3)
Solution: Split into two constraints
𝑥+𝑦≥3
𝑥+𝑦≤3

12. Converting to Canonical Form
Free variable: x may or may not be nonnegative.
Solution: Split x into x1 and x2
𝑥= 𝑥 1 − 𝑥 2 𝑥 1 ≥0 𝑥 2 ≥0

13. Using LP to solve graph problems
Edge (i, j): Course i can be scheduled into classroom j
Courses
Classrooms
Solution: A set of edges that do not share any vertices. (a matching)

14. Using LP to solve graph problems1 2 1 2 3 3 1 2 4 5 6 4 1 1 3 2 7 8 9 1 1