Lecture 04-05-17

Review:

Recall: steps to show X is NP-complete
1. Show X is in NP
  - Verify that the answer is yes
2. Show that a known NP-complete problem Y is polynomially reducable to X
  - Show a clear reduction
  - If the answer to Y is yes, then the answer to transformed problem is yes
  - If the answer to X is yes, then the answer to original problem, Y, is yes
NP-complete:
- Must be in NP
  - Decision problem where you can verify the answer is yes in polynomial time
NP:
1. Verify the answer in polynomial time
2. Solved in polynomial time by a nondeterministic Turing machine

Bin packing
- Usually teach the easiest version that is still hard
- Easiest thing to pack: 1d numbers
- Given numbers, want to pack in least number of bins possible
- Decision problem: given items, is it possible to pack it into k bins
Knapsack problem
- Objects with weight, knapsack has capacity
- Come as close to filling the knapsack as possible
- Can you put objects in whose weight sums to C?
Coloring problem
- In a graph, if vertices are adjacent, then they must get different colors
- Make each of the areas a vertex
- Famous question: can all maps be colored with 4 colors?
  - Planar graphs
    - No edges crossing
- Application:
  - Conflict graphs
- Given graph G, can it be colored with K numbers?
  - chormatic number is the minimum number of colors it takes
Traveling salesperson
- Version: G, K
  - G is a weighted graph and complete
    - Is an edge between each pair of vertices
    - Simplest version
- Is there a tour of cost <= K?
- Is there a cycle that goes through every vertex exactly once and has total cost <= K?
- Other versions
  - Wandering salesperson problem