Lecture 23 Intractable Problems Smiley Puzzles and Curing

Lecture 23: Intractable Problems (Smiley Puzzles and Curing Cancer) CS 200: Computer Science David Evans University of Virginia 21 March 2003 CS 200 Spring 2003 http: //www. cs. virginia. edu/evans Computer Science

Menu • Review: P and NP • NP Problems • NP-Complete Problems 21 March 2003 CS 200 Spring 2003 2

Complexity 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. 21 March 2003 CS 200 Spring 2003 3

Complexity 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. 21 March 2003 CS 200 Spring 2003 4

Why O(2 n) work is “intractable” time since “Big Bang” n! 2 n 2022 today 21 March 2003 n 2 n log n CS 200 Spring 2003 log-log scale 5

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”. 21 March 2003 CS 200 Spring 2003 6

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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! 21 March 2003 CS 200 Spring 2003 8

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) 21 March 2003 CS 200 Spring 2003 9

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. 21 March 2003 CS 200 Spring 2003 10

This makes a huge difference! time since “Big Bang” n! 2 n Solving the 5 x 5 smileys problem either takes a few seconds, or more time than the universe has been in existence. But, no one knows 2032 which for sure! today 21 March 2003 CS 200 Spring 2003 n 2 n log-log scale 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 21 March 2003 CS 200 Spring 2003 12

3 SAT Smiley 21 March 2003 CS 200 Spring 2003 Step 1: Transform into smileys Step 2: Solve (using our fast smiley puzzle solving procedure) Step 3: Invert transform (back into 3 SAT problem 13

The Real 3 SAT Problem (also can be quickly transformed into the Smileys Puzzle) 21 March 2003 CS 200 Spring 2003 14

Propositional Grammar Sentence : : = Clause Sentence Rule: Evaluates to value of 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 21 March 2003 CS 200 Spring 2003 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 Rule: Evaluates to value associated with Name. 21 March 2003 CS 200 Spring 2003 16

Proposition Example Sentence : : = Clause 1 Clause 2 Clause : : = ( Clause ) Clause : : = Name (or) (and) (not) a (b c) b c 21 March 2003 CS 200 Spring 2003 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) 21 March 2003 CS 200 Spring 2003 18

SAT Example Sentence : : = Clause 1 Clause 2 Clause : : = ( Clause ) Clause : : = Name (or) (and) (not) SAT (a (b c) { a: true, b: false, c: true } { a: true, b: true, c: false } SAT (a a) no way 21 March 2003 CS 200 Spring 2003 19

The 3 SAT 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) 21 March 2003 CS 200 Spring 2003 20

3 SAT / SAT Is 3 SAT easier or harder than SAT? It is definitely not harder than SAT, since all 3 SAT problems are also SAT problems. Some SAT problems are not 3 SAT problems. 21 March 2003 CS 200 Spring 2003 21

3 SAT Example Sentence : : = Clause 1 Clause 2 Clause : : = ( Clause ) Clause : : = Name (or) (and) (not) 3 SAT ( (a b c) ( a b d) ( a b d) (b c d ) ) { a: true, b: false, c: false, d: false} 21 March 2003 CS 200 Spring 2003 22

3 SAT Smiley • Like 3/stone/apple/tower puzzle, we can convert every 3 SAT 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 3 SAT also! 21 March 2003 CS 200 Spring 2003 23

NP Complete • Cook and Levin proved that 3 SAT was NP -Complete (1971) • A problem is NP-complete if it is as hard as the hardest problem in NP • If 3 SAT 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! 21 March 2003 CS 200 Spring 2003 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) 21 March 2003 CS 200 Spring 2003 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 traveled 21 March 2003 CS 200 Spring 2003 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. 21 March 2003 CS 200 Spring 2003 27

Phylogeny Problem – Input: a set of n species and their genomes – Output: a tree that connects all the species in a way that requires the less than k total mutations or “impossible” if there is no such tree. 21 March 2003 CS 200 Spring 2003 28

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. 21 March 2003 CS 200 Spring 2003 29

Perfect 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”. Note: not a perfect photomosaic! If given a tiling, easy to check if the total color difference is less than the minimum. 21 March 2003 CS 200 Spring 2003 30

Drug Discovery Problem – Input: a set of proteins, a desired 3 D 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 = $200 B/year 21 March 2003 CS 200 Spring 2003 31

Is it ever useful to be confident that a problem is hard? 21 March 2003 CS 200 Spring 2003 32

Factoring Problem • • – Input: an n-digit number – Output: two prime factors whose product is the input number Easy to multiply to check factors are correct Not proven to be NP-Complete (but probably is) Most used public key cryptosystem (RSA) depends on this being hard See “Sneakers” for what solving this means 21 March 2003 CS 200 Spring 2003 33

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 ~$1 Trillion) – 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 ~$1 M) • Most computer scientists think they are intractable…but no one knows for sure! 21 March 2003 CS 200 Spring 2003 34

Speculations • Must study math for 15 years before understanding an open problem – Was ~10 until Andrew Wiles proved Fermat’s Last Theorem • Must study physics for ~6 years before understanding an open problem • Must study computer science for ~6 weeks before understanding the most important open problem – Unless you’re a 6 -year old at Cracker Barrel • But, every 5 year-old understands the most important open problems in biology! 21 March 2003 CS 200 Spring 2003 35

Charge • Problem Set 6: Due Monday • Minesweeper is NP complete (optional) • We still aren’t sure if the Cracker Barrel Puzzle is NP-Complete – Chris Frost and Mike Peck are working on it (except Chris is off in Beverly Hills, see Daily Progress article) – Will present about their proof at a later lecture 21 March 2003 CS 200 Spring 2003 36