CS200: Computer Science, Spring 2004

Notes: Wednesday 21 April 2004
Schedule
 Monday, 26 April (last day of class): PS8
 Friday, 30 April, 4:55pm: Final Exam (handed out 26 April)
Notes What is the difference between a tractable and intractable problem?
What does it mean for a problem to be NPcomplete?
What are the characteristics of NPcomplete problems?
Problem Classification Arguments To show a problem is decidable/in NP/in P, you need to show it is easy enough to be solved with a procedure in that class:
 Decidable: it is easy enough to be solved by some procedure that always terminates
 NP: it is easy enough to be solved by a procedure that tries an exponential number of guesses, but takes only Ptime to check one if correct
 P: it is easy enough to be solved by a polynomial time procedure (that is, a procedure that is O(n^{k}).)
To show a problem is undecidable or NPcomplete, you need to show it is as hard as a problem you already know is in that class:
How would you convince someone the smiley puzzle is NPcomplete?
 Undecidable: if you had a terminating procedure that solves this problem, you could use it to solve a known undecidable problem (e.g., the halting problem).
 NPComplete: if you had a procedure that solves this problem, you could use it to solve a known NPComplete problem (e.g., 3SAT). Subtlety: the transformation of the problem and answer must be in P.
How would it impact medicine if you discovered a Θ(n^{7}) procedure that solves the smiley puzzle?
How would it impact electronic commerce if you discovered a Θ(n^{7}) procedure that solves the smiley puzzle?
How would it impact electronic commerce if you discovered a Θ(5^{n}) procedure that solves the smiley puzzle?
Is the smiley puzzle harder than the Cracker Barrel Peg Board puzzle?
LinksP vs. NP, Clay Mathematics Institute, Millennium Prize Problem.
Minesweeper, Ian Stewart. (The Minesweeper Consistency Problem is NPComplete!)
Phylogeny (CS201J Problem Set)Christopher Frost, Michael Peck, David Evans. Pancakes, Puzzles, and Polynomials: Cracking the Cracker Barrel. ACM Special Interest Group on Algorithms and Computation Theory (SIGACT) News, March 2004. This paper (attached) shows how to prove a variant of the pegboard puzzle is NPComplete by showing that if you could solve the pegboard puzzle in polynomial time you could also solve 3SAT. After publishing it, we learned from a reader that there is also a proof that the original generalized pegboard puzzle (HiQ) is also NPComplete. You can see slides from Chris and Mike's URDS presentation here: http://www.cs.virginia.edu/pegboard/cburds.ppt.
cs200staff@cs.virginia.edu Using these Materials 