David Evans http://www.cs.virginia.edu/evans Class 14: Intractable Problems CS150: Computer Science University of Virginia Computer Science David Evans http://www.cs.virginia.edu/evans

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!

Problems and Procedures To know a O (f) bound for a problem, we need to find a (f) procedure that solves it The sorting problem is O (n log n) since we know a procedure that solves it in (n log n) To know a Ω(f ) bound for a problem, we need to prove that there is no procedure faster than (f) that solves it We proved sorting is Ω(n log n) by reasoning about the number of decisions needed

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!

Complexity Class P “Tractable” 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, simulating the universe are all in P.

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. Alternately: If we had a magic guess-correctly procedure that makes every decision correctly, we could devise a procedure that solves the problem in polynomial time. Note: this definition is not precise enough to be satisfying yet! We will need to understand better what a “step” means when we measure work to define this properly.

NP Problems The smileys problem is in NP 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)

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.

Orders of Growth 2n < n! simulating universe smileys puzzle insertsort quicksort

Orders of Growth smileys puzzle simulating universe insertsort quicksort

Orders of Growth Smileys puzzle “intractable” “tractable” simulating universe I do nothing that a man of unlimited funds, superb physical endurance, and maximum scientific knowledge could not do. – Batman (may be able to solve intractable problems, but computer scientists can only solve tractable ones for large n)

Quiz Break

Intractable Problems P n! 2n n2 n log n log-log scale time since “Big Bang” 2n P 2022 today n2 n log n

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

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)

This makes a huge difference! time since “Big Bang” 2n 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

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

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    

The Real 3SAT Problem (also can be quickly transformed into the Smileys Puzzle)

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

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.

a  (b  c)  b  c Proposition Example Sentence ::= Clause Clause ::= Clause1  Clause2 (or) Clause ::= Clause1  Clause2 (and) Clause ::= Clause (not) Clause ::= ( Clause ) Clause ::= Name Proposition Example a  (b  c)  b  c

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)

SAT Example SAT (a  (b  c)  b  c) SAT (a  a) 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

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)

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.

3SAT Example 3SAT ( (a  b   c)  (a   b  d)  (a  b   d) 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}

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!

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!

Charge PS4 is due Monday More on P vs. NP next class