1. Week 10 - Monday
CS 113

2. Last time What did we talk about last time?
Reading (and writing, sort of) files in Python
Lab 9

3. Questions?

4. Project 3

5. Hard Problems

6. Hamiltonian Cycle
The Eulerian Cycle problem asked if you could start at some node, cross every edge once, and return to the start
The Hamiltonian Cycle problem asks if you can start at some node, visit every node only once, and return to the start
In other words, find a tour
Sounds easy, right?

7. Find the Hamiltonian Cycle
On this graph: A D B E C D A E B C

8. Find the Hamiltonian Cycle
Now, on this graph:
There isn’t one!

9. Find the Hamiltonian Cycle
Now, on this graph:
Hmmm…

10. Traveling Salesman Problem: Hamiltonian Cycle Meets Shortest Path
Given a graph, find the shortest tour that visits every node and comes back to the starting point
Like a lazy traveling salesman who wants to visit all possible clients and then return to his home

11. What’s the shortest tour?
Find the shortest TSP tour:
Starts anywhere
Visits all nodes
Has the shortest length of such a tour A 5 6 4 3 2 3 E B D 3 F 7 C

12. How can we always find the shortest tour?
Strategies:
Always add the nearest neighbor
Randomized approaches
Try every possible combination
Why are we only listing strategies?

13. Best Solution to TSP
So far we haven’t given a foolproof solution to TSP
Let’s look at two possibilities:
Greedy Solution: Pick the closest neighbor
Brute Force: Try all possibilities

14. Greedy Doesn’t Work
We are tempted to always take the closest neighbor, but there are pitfalls
Greedy
Optimal

15. Brute Force is Brutal
In a completely connected graph, we can try any sequence of nodes
If there are n nodes, there are (n – 1)! tours
For 30 cities, 29! =
If we can check 1,000,000,000 tours in one second, it will only take about 20,000 times the age of the universe to check them all
We will (eventually) get the best answer!

16. Running time How do we measure the amount of time an algorithm takes?
Surely running TSP on 4 cities takes less time than running TSP on 1,000 cities
1,000 cities in a straight line seems easier than 1,000 cities randomly scattered
We use Big Oh notation!

17. What’s the best time for TSP?
No one knows how to find the best solution to TSP in an efficient amount of time
The best solutions known take O(2n)
For a general graph, no good approximation even exists
For a graph with the triangle inequality, there is an approximation that yields a tour no more than 3/2 the optimal
Some variations on the problem are easier than others

18. Hard problems Is TSP the hardest problem there is?
Are there other problems equally hard?
How do we compare the difficulty of one problem to another?

19. NP-completeness
TSP is one of many problems which are called NP-complete
All of these problems share the characteristic that the only way we know to find best solution uses brute force
All NP-complete problems are reducible to all other NP-complete problems
An efficient solution to one would guarantee an efficient solution to all

20. NP-complete problems on graphs
Traveling Salesman Problem
Hamiltonian Cycle (and Path)
Let’s see just a couple more…

21. Graph Coloring
Find the smallest number of colors for coloring nodes such that no two adjacent nodes have the same color E A D B F C A E B D F C

22. Graph Coloring
What about this graph?

23. Graph Coloring
Like TSP and Hamiltonian Cycle, large graphs get really hard to find the smallest coloring for
It might not be obvious, but graph coloring has practical applications
Perhaps the most common example is register allocation inside of a compiler

24. Maximum Clique Find the largest complete subgraph in a given graph
This graph has a clique of size 3

25. Maximum Clique What about this graph?
This graph has a clique of size 4
As before, a large graph can be very difficult

26. Knapsack problem Not all NP-complete problems are graph problems
The knapsack problem is the following:
Imagine that you are Indiana Jones
You are the first to open the tomb of some long-lost pharaoh
You have a knapsack that can hold m pounds of loot, but there's way more than that in the tomb
Because you're Indiana Jones, you can instantly tell how much everything weighs and how valuable it is
You want to find the most valuable loot that weighs less than or equal to m pounds

27. Subset sum This one is a little bit mathematical
Say you have a set of numbers
Somebody gives you a number k
Is there any subset of the numbers in your set that add up to exactly k?
Example:
Set: { 3, 9, 15, 22, 35, 52, 78, 141}
Is there a subset that adds up to exactly 100?
What about 101?

28. NP NP is actually a class of problem
Problems in NP can be solved in polynomial time on a non-deterministic computer
A deterministic computer is the kind you know:
First it has to consider possibility A, then, it can consider possibility B

29. Deterministic vs. non-deterministic
A non-deterministic computer (which, as far as we know, only exists in our imagination) can consider both possibility A and possibility B at the same time
Deterministic A B C D E F A B C D E F
Non-Deterministic

30. P
P is the class of decision problems that can be solved in polynomial time by a deterministic computer
Lots of great problems are in P:
Is this list sorted?
Is this number prime?
Is the shortest path from node A to node B of length 37?
It's unknown whether some problems are in P:
Does this number have exactly two factors?
Is this graph equivalent to this other graph?

31. NP-complete Everything in P is also in NP
Some problems are the “hardest” problems in NP
This means that any problem in NP can be converted into one of these problem in polynomial time
These problems make up the class NP-complete
NP-complete
NP-complete

32. Decisions, decisions
Notice that P, NP, and NP-complete are all decision problems
So, the TSP we stated is not technically NP-complete
The NP-complete version is:
Is there a tour of length less than or equal to 24 in this graph?
The optimization versions of NP-complete problems are called NP-hard

33. Easy to check vs. easy to answer
Computer scientists view problems in P as "easy to answer"
They can be computed in polynomial time
Problems in NP are "easy to check"
An answer can be checked in polynomial time
For example, if someone gives you a Traveling Salesman tour, you can verify that it is a legal tour of the required length
But is easy to check the same as easy to answer?

34. Fabulous cash prizes These NP-complete problems are very hard
Many of them are really useful
Clay Mathematics Institute has offered a $1,000,000 prize
You can do one of two things to collect it:
Find an efficient solution to any of the problems
Prove that one cannot have an efficient solution

35. What if P = NP? Most computer scientists think that P ≠ NP
But if it were
Most things could be perfectly scheduled
e.g., the best room for a given number of students and the time preferences of everyone involved
All routing and path planning (UPS, military, etc.) would be optimal
It might be possible to devise perfect genetic therapies for certain conditions
It would be possible to prove all kinds of previously unproven theorems in mathematics

36. What if P = NP? (The Bad)
On the other hand, if P = NP, it might also mean:
Most of our encryption algorithms would be broken
All computer, Internet, and banking security would be worthless
Could creativity be doomed?
If recognizing something good was the same as creating something good… who knows?

37. A final word from an MIT professor
If P=NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in “creative leaps,” no fundamental gap between solving a problem and recognizing the solution once it’s found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss; everyone who could recognize a good investment strategy would be Warren Buffett. It’s possible to put the point in Darwinian terms: if this is the sort of universe we inhabited, why wouldn’t we already have evolved to take advantage of it? Scott Aaronson

38. Upcoming

39. Next time…
Finish NP-completeness
Turing machines
Halting problem

40. Reminders Design document for final project is due Friday
Start working on Project 4
Exam 2 is next Monday
We'll review on Friday
Talk: Creativity in the Many Areas of Game Development at Firaxis Games
Barry Caudill
Wednesday 11am-12:25 in E273