David Evanshttp://www.cs.virginia.edu/evans

CS200: Computer ScienceUniversity of VirginiaComputer Science

Class 38:Intractable Problems(Smiley Puzzles and Curing Cancer)

21 April 2004 CS 200 Spring 2004 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

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Complexity ClassesClass 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).

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Problem Classes

PNP

Decidable

Undecidable

Sorting: (n log n)

Simulating Universe: O(n3)

Cracker Barrel: O(2n) and (n)

Fill tape with 2n *s: (2n)

Halting Problem: ()

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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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If P NP:

PNP

Decidable

Undecidable

Sorting: (n log n)

Simulating Universe: O(n3)

Cracker Barrel: O(2n) and (n)

Fill tape with 2n *s: (2n)

Halting Problem: ()

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If P = NP:

PNP

Decidable

Undecidable

Sorting: (n log n)

Simulating Universe: O(n3)

Cracker Barrel: O(2n) and (n)

Fill tape with 2n *s: (2n)

Halting Problem: ()

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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”.

Thanks to Peggy Reed for making the Smiley Puzzles!

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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!

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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 NPWe can easily try n! different answers

We can quickly check if a guess is correct (check all n tiles)

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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.

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This makes a huge difference!

1

100

10000

1E+06

1E+08

1E+10

1E+12

1E+14

1E+16

1E+18

1E+20

1E+22

1E+24

1E+26

1E+28

1E+30

2 4 8 16 32 64 128

n! 2n

n2

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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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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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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The Real 3SAT Problem(also can be quickly transformed

into the Smileys Puzzle)

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Propositional GrammarSentence ::= 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

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Propositional GrammarClause ::= Clause

Not Rule: Evaluates to the opposite value of clause (true false)

Clause ::= ( Clause )Group Rule: Evaluates to value of clause.

Clause ::= NameName Rule: Evaluates to value associated with Name.

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PropositionExample

Sentence ::= Clause

Clause ::= Clause1 Clause2 (or)

Clause ::= Clause1 Clause2 (and)

Clause ::= Clause (not)

Clause ::= ( Clause )

Clause ::= Name

a (b c) b c

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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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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 ::= Clause1 Clause2 (or)

Clause ::= Clause1 Clause2 (and)

Clause ::= Clause (not)

Clause ::= ( Clause )

Clause ::= Name

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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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3SAT / SAT

Is 3SAT easier or harder than SAT?

It is definitely not harder than SAT, since all 3SAT problemsare also SAT problems. Some SAT problems are not 3SAT problems.

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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 ::= Clause1 Clause2 (or)

Clause ::= Clause1 Clause2 (and)

Clause ::= Clause (not)

Clause ::= ( Clause )

Clause ::= Name

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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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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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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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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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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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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 isattached to today’s notes.

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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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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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Is it ever useful to be confident that a problem is

hard?