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

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21 April 2004CS 200 Spring 20042 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

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21 April 2004CS 200 Spring 20043 Complexity Classes Class P: problems that can be solved in polynomial time by a deterministic TM. O (n k ) 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).

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21 April 2004CS 200 Spring 20044 Problem Classes P NP Decidable Undecidable Sorting: (n log n) Simulating Universe: O(n 3 ) Cracker Barrel: O(2 n ) and (n) Fill tape with 2 n *s: (2 n ) Halting Problem: ( )

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21 April 2004CS 200 Spring 20045 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)

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21 April 2004CS 200 Spring 20046 If P NP: P NP Decidable Undecidable Sorting: (n log n) Simulating Universe: O(n 3 ) Cracker Barrel: O(2 n ) and (n) Fill tape with 2 n *s: (2 n ) Halting Problem: ( )

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21 April 2004CS 200 Spring 20047 If P = NP: P NP Decidable Undecidable Sorting: (n log n) Simulating Universe: O(n 3 ) Cracker Barrel: O(2 n ) and (n) Fill tape with 2 n *s: (2 n ) Halting Problem: ( )

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21 April 2004CS 200 Spring 20048 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”.

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Thanks to Peggy Reed for making the Smiley Puzzles!

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21 April 2004CS 200 Spring 200410 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: ( ) N o one knows!

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21 April 2004CS 200 Spring 200411 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)

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21 April 2004CS 200 Spring 200412 Is the Smiley’s Problem in P? No one knows! We can’t find a O( n k ) solution. We can’t prove one doesn’t exist.

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21 April 2004CS 200 Spring 200413 This makes a huge difference! n! 2n2n n2n2 n log n today 2032 time since “Big Bang” log-log scale 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!

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21 April 2004CS 200 Spring 200414 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

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21 April 2004CS 200 Spring 200415 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

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21 April 2004CS 200 Spring 200416 The Real 3SAT Problem (also can be quickly transformed into the Smileys Puzzle)

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21 April 2004CS 200 Spring 200417 Propositional Grammar Sentence ::= Clause Sentence Rule: Evaluates to value of Clause Clause ::= Clause 1 Clause 2 Or Rule: Evaluates to true if either clause is true Clause ::= Clause 1 Clause 2 And Rule: Evaluates to true iff both clauses are true

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21 April 2004CS 200 Spring 200418 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.

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21 April 2004CS 200 Spring 200419 Proposition Example Sentence ::= Clause Clause ::= Clause 1 Clause 2 (or) Clause ::= Clause 1 Clause 2 (and) Clause ::= Clause (not) Clause ::= ( Clause ) Clause ::= Name a (b c) b c

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21 April 2004CS 200 Spring 200420 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)

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21 April 2004CS 200 Spring 200421 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 Sentence ::= Clause Clause ::= Clause 1 Clause 2 (or) Clause ::= Clause 1 Clause 2 (and) Clause ::= Clause (not) Clause ::= ( Clause ) Clause ::= Name

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21 April 2004CS 200 Spring 200422 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)

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21 April 2004CS 200 Spring 200423 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.

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21 April 2004CS 200 Spring 200424 3SAT Example 3SAT ( (a b c) ( a b d) ( a b d) (b c d ) ) { a: true, b: false, c: false, d: false } Sentence ::= Clause Clause ::= Clause 1 Clause 2 (or) Clause ::= Clause 1 Clause 2 (and) Clause ::= Clause (not) Clause ::= ( Clause ) Clause ::= Name

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21 April 2004CS 200 Spring 200425 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!

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21 April 2004CS 200 Spring 200426 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!

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21 April 2004CS 200 Spring 200427 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)

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21 April 2004CS 200 Spring 200428 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

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21 April 2004CS 200 Spring 200429 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.

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21 April 2004CS 200 Spring 200430 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.

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21 April 2004CS 200 Spring 200431 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.

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21 April 2004CS 200 Spring 200432 Drug Discovery Problem –Input: a set of proteins, a desired 3D shape –Output: a sequence of proteins that produces the shape (or impossible) Note: US Drug sales = $200B/year If given a sequence, easy (not really) to check if sequence has the right shape. Caffeine

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21 April 2004CS 200 Spring 200433 Is it ever useful to be confident that a problem is hard?

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