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David Evans cs.virginia/evans

Class 38: Intractable Problems (Smiley Puzzles and Curing Cancer). David Evans http://www.cs.virginia.edu/evans. CS200: Computer Science University of Virginia Computer Science. Complexity and Computability. We’ve learned how to measure the complexity of any procedure ( )

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David Evans cs.virginia/evans

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  1. Class 38: Intractable Problems (Smiley Puzzles and Curing Cancer) David Evans http://www.cs.virginia.edu/evans CS200: Computer Science University of Virginia Computer Science

  2. Complexity and Computability • We’ve learned how to measure the complexity of any procedure () • Average score on question 6 (8.9/10), 7 (8.6/10) and 8 (8.1/10) • We’ve learned how to show some problems are undecidable • Harder (average on question 4 = 7.7) • Today: reasoning about the complexity of some problems CS 200 Spring 2004

  3. Complexity Classes Class P: problems that can be solved in polynomial time by a deterministic TM. O (nk) for some constant k. Easy problems like simulating the universe are all in P. Class NP: problems that can be solved in polynomial time by a nondeterministic TM Hard problems like the pegboard puzzle sorting are in NP (as well as all problems in P). CS 200 Spring 2004

  4. Problem Classes Fill tape with 2n *s: (2n) Simulating Universe: O(n3) Undecidable NP Decidable P Halting Problem: () Sorting: (n log n) Cracker Barrel: O(2n) and (n) CS 200 Spring 2004

  5. P = NP? • Is there a polynomial-time solution to the “hardest” problems in NP? • No one knows the answer! • The most famous unsolved problem in computer science and math • Listed first on Millennium Prize Problems • win $1M if you can solve it • (also an automatic A+ in this course) CS 200 Spring 2004

  6. If P  NP: Fill tape with 2n *s: (2n) Simulating Universe: O(n3) Undecidable NP Decidable P Halting Problem: () Sorting: (n log n) Cracker Barrel: O(2n) and (n) CS 200 Spring 2004

  7. If P = NP: Fill tape with 2n *s: (2n) Simulating Universe: O(n3) Undecidable NP Decidable P Halting Problem: () Sorting: (n log n) Cracker Barrel: O(2n) and (n) CS 200 Spring 2004

  8. Smileys Problem Input: n square tiles Output: Arrangement of the tiles in a square, where the colors and shapes match up, or “no, its impossible”. CS 200 Spring 2004

  9. Thanks to Peggy Reed for making the Smiley Puzzles!

  10. How much work is the Smiley’s Problem? • Upper bound: (O) O (n!) Try all possible permutations • Lower bound: ()  (n) Must at least look at every tile • Tight bound: () No one knows! CS 200 Spring 2004

  11. NP Problems • Can be solved by just trying all possible answers until we find one that is right • Easy to quickly check if an answer is right • Checking an answer is in P • The smileys problem is in NP We can easily try n! different answers We can quickly check if a guess is correct (check all n tiles) CS 200 Spring 2004

  12. Is the Smiley’s Problem in P? No one knows! We can’t find a O(nk) solution. We can’t prove one doesn’t exist. CS 200 Spring 2004

  13. This makes a huge difference! n! 2n time since “Big Bang” Solving a large smileys problem either takes a few seconds, or more time than the universe has been in existence. But, no one knows which for sure! 2032 today n2 n log n log-log scale CS 200 Spring 2004

  14. Who cares about Smiley puzzles? If we had a fast (polynomial time) procedure to solve the smiley puzzle, we would also have a fast procedure to solve the 3/stone/apple/tower puzzle: 3 CS 200 Spring 2004

  15. 3SAT  Smiley Step 1: Transform into smileys Step 2: Solve (using our fast smiley puzzle solving procedure) Step 3: Invert transform (back into 3SAT problem     CS 200 Spring 2004

  16. The Real 3SAT Problem(also can be quickly transformed into the Smileys Puzzle) CS 200 Spring 2004

  17. Propositional Grammar Sentence ::= Clause Sentence Rule:Evaluates to value of Clause Clause ::= Clause1 Clause2 Or Rule:Evaluates to true if either clause is true Clause ::= Clause1Clause2 And Rule:Evaluates to true iff both clauses are true CS 200 Spring 2004

  18. Propositional Grammar Clause ::= Clause Not Rule:Evaluates to the opposite value of clause (truefalse) Clause ::= ( Clause ) Group Rule:Evaluates to value of clause. Clause ::= Name Name Rule:Evaluates to value associated with Name. CS 200 Spring 2004

  19. PropositionExample Sentence ::= Clause Clause ::= Clause1 Clause2 (or) Clause ::= Clause1Clause2 (and) Clause ::= Clause (not) Clause ::= ( Clause ) Clause ::= Name a  (b  c)  b  c CS 200 Spring 2004

  20. The Satisfiability Problem (SAT) • Input: a sentence in propositional grammar • Output: Either a mapping from names to values that satisfies the input sentence or no way (meaning there is no possible assignment that satisfies the input sentence) CS 200 Spring 2004

  21. Sentence ::= Clause Clause ::= Clause1 Clause2 (or) Clause ::= Clause1Clause2 (and) Clause ::= Clause (not) Clause ::= ( Clause ) Clause ::= Name SAT Example SAT (a  (b  c)  b  c)  { a:true, b: false,c:true }  { a:true, b: true,c:false } SAT (a  a)  no way CS 200 Spring 2004

  22. The 3SAT Problem • Input: a sentence in propositional grammar, where each clause is a disjunction of 3 names which may be negated. • Output: Either a mapping from names to values that satisfies the input sentence or no way (meaning there is no possible assignment that satisfies the input sentence) CS 200 Spring 2004

  23. 3SAT / SAT Is 3SAT easier or harder than SAT? It is definitely not harder than SAT, since all 3SAT problems are also SAT problems. Some SAT problems are not 3SAT problems. CS 200 Spring 2004

  24. Sentence ::= Clause Clause ::= Clause1 Clause2 (or) Clause ::= Clause1Clause2 (and) Clause ::= Clause (not) Clause ::= ( Clause ) Clause ::= Name 3SAT Example 3SAT ( (a  b   c)  (a   b  d)  (a  b   d)  (b   c  d ) )  { a:true, b: false,c:false,d:false} CS 200 Spring 2004

  25. 3SAT  Smiley • Like 3/stone/apple/tower puzzle, we can convert every 3SAT problem into a Smiley Puzzle problem! • Transformation is more complicated, but still polynomial time. • So, if we have a fast (P) solution to Smiley Puzzle, we have a fast solution to 3SAT also! CS 200 Spring 2004

  26. NP Complete • Cook and Levin proved that 3SAT was NP-Complete (1971) • A problem is NP-complete if it is as hard as the hardest problem in NP • If 3SAT can be transformed into a different problem in polynomial time, than that problem must also be NP-complete. • Either all NP-complete problems are tractable (in P) or none of them are! CS 200 Spring 2004

  27. NP-Complete Problems • Easy way to solve by trying all possible guesses • If given the “yes” answer, quick (in P) way to check if it is right • Solution to puzzle (see if it looks right) • Assignments of values to names (evaluate logical proposition in linear time) • If given the “no” answer, no quick way to check if it is right • No solution (can’t tell there isn’t one) • No way (can’t tell there isn’t one) CS 200 Spring 2004

  28. Traveling Salesperson Problem • Input: a graph of cities and roads with distance connecting them and a minimum total distant • Output: either a path that visits each with a cost less than the minimum, or “no”. • If given a path, easy to check if it visits every city with less than minimum distance traveled CS 200 Spring 2004

  29. Graph Coloring Problem • Input: a graph of nodes with edges connecting them and a minimum number of colors • Output: either a coloring of the nodes such that no connected nodes have the same color, or “no”. If given a coloring, easy to check if it no connected nodes have the same color, and the number of colors used. CS 200 Spring 2004

  30. Pegboard Problem - Input: a configuration of n pegs on a cracker barrel style pegboard - Output: if there is a sequence of jumps that leaves a single peg, output that sequence of jumps. Otherwise, output false. If given the sequence of jumps, easy (O(n)) to check it is correct. If not, hard to know if there is a solution. Proof that variant of this problem is NP-Complete is attached to today’s notes. CS 200 Spring 2004

  31. Minesweeper Consistency Problem • Input: a position of n squares in the game Minesweeper • Output: either a assignment of bombs to squares, or “no”. • If given a bomb assignment, easy to check if it is consistent. CS 200 Spring 2004

  32. Drug Discovery Problem • Input: a set of proteins, a desired 3D shape • Output: a sequence of proteins that produces the shape (or impossible) Caffeine If given a sequence, easy (not really) to check if sequence has the right shape. Note: US Drug sales = $200B/year CS 200 Spring 2004

  33. Is it ever useful to be confident that a problem is hard? CS 200 Spring 2004

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