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David Evans cs.virginia/~evans

Lecture 15: Intractable Problems (Smiley Puzzles and Curing Cancer). David Evans http://www.cs.virginia.edu/~evans. CS200: Computer Science University of Virginia Computer Science. Menu. P and NP NP Problems NP-complete Problems. Complexity Class P.

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David Evans cs.virginia/~evans

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  1. Lecture 15: Intractable Problems (Smiley Puzzles and Curing Cancer) David Evans http://www.cs.virginia.edu/~evans CS200: Computer Science University of Virginia Computer Science

  2. Menu • P and NP • NP Problems • NP-complete Problems CS 200 Spring 2002

  3. Complexity Class P Class P: problems that can be solved in polynomial time O(nk) for some constant k. Easy problems like sorting, making a photomosaic using duplicate tiles, understanding the universe. CS 200 Spring 2002

  4. Complexity Class NP Class NP: problems that can be solved in nondeterministic polynomial time If we could try all possible solutions at once, we could identify the solution in polynomial time. We’ll see a few examples today. CS 200 Spring 2002

  5. Growth Rates n2 (bubblesort) n log2n (quicksort) CS 200 Spring 2002

  6. Why O(2n) work is “intractable” n! 2n time since “Big Bang” 2022 today n2 n log n log-log scale CS 200 Spring 2002

  7. Moore’s Law Doesn’t Help • If the fastest procedure to solve a problem is O(2n) or worse, Moore’s Law doesn’t help much. • Every doubling in computing power increases the problem size by 1. CS 200 Spring 2002

  8. Smileys Problem Input: 16 square tiles Output: Arrangement of the tiles in 4x4 square, where the colors and shapes match up, or “no, its impossible”. CS 200 Spring 2002

  9. NP Problems • Best way we know to solve it is to try all possible permutations until we find one that is right • Easy to check if an answer is right • Just look at all the edges if they match • We know the simleys problem is: O(n!) and (n) but no one knows if it is (n!) or (n) CS 200 Spring 2002

  10. This makes a huge difference! n! 2n time since “Big Bang” Solving the 5x5 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 2002

  11. 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 2002

  12. 3SAT  Smiley • Step 1: Transform   Step 2: Solve (using our fast smiley puzzle solving procedure) Step 3: Invert transform (back into 3SAT problem   CS 200 Spring 2002

  13. The Real 3SAT Problem(also identical to Smileys Puzzle) CS 200 Spring 2002

  14. 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 2002

  15. 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 2002

  16. 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 2002

  17. 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 2002

  18. 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 2002

  19. 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 2002

  20. 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 2002

  21. 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 2002

  22. 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 2002

  23. NP Complete • 3SAT and Smiley Puzzle are examples of NP-complete Problems • A problem is NP-complete if it is as hard as the hardest problem in NP • All NP problems can be mapped to any of the NP-complete problems in polynomial time • Either all NP-complete problems are tractable (in P) or none of them are! CS 200 Spring 2002

  24. 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 2002

  25. Traveling Salesman 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 travelled CS 200 Spring 2002

  26. 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 2002

  27. Minesweeper Consistency Problem • Input: a position in the game Minesweeper • Output: either a assignment of bombs to variables, or “no”. • If given a bomb assignment, easy to check if it is consistent. CS 200 Spring 2002

  28. Photomosaic Problem • Input: a set of tiles, a master image, a color difference function, and a minimum total difference • Output: either a tiling with total color difference less than the minimum or “no”. • If given a tiling, easy to check if the total color difference is less than the minimum. CS 200 Spring 2002

  29. Drug Discovery Problem • Input: a set of proteins, a desired 3D shape • Output: a sequence of proteins that produces the shape • If given a sequence, easy (not really) to check if sequence has the right shape. Note: solving the minesweeper problem may be worth $1M, but if you can solve this one its worth $1Trillion+. (US Drug sales = $200B/year) CS 200 Spring 2002

  30. Factoring Problem • Input: an n-digit number • Output: a set of factors whose product is the input number • Given the factors, easy to multiply to check if they are correct • Not proven to be NP-Complete (but probably is) • See “Sneakers” for what solving this one is worth… (PS7 will consider this) CS 200 Spring 2002

  31. NP-Complete Problems • These problems sound really different…but they are really the same! • A P solution to any NP-Complete problem makes all of them in P (worth ~$1Trillion) • Solving the minesweeper constraint problem is actually as good as solving drug discovery! • A proof that any one of them has no polynomial time solution means none of them have polynomial time solutions (worth ~$1M) • Most computer scientists think they are intractable…but no one knows for sure! CS 200 Spring 2002

  32. Charge • Later in the course: • There are some problems even harder than NP complete problems! • There are problems we can prove are not solvable, no matter how much time you have. (Gödel – your spring break reading) • PS4 Due Friday (Lab Hours Thu 7-9pm) • Friday • Either review for exam or reduction example CS 200 Spring 2002

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