1 Polynomial Church-Turing thesis A decision problem can be solved in polynomial time by using a reasonable sequential model of computation if and only if it can be solved in polynomial time by a Turing machine.

2 The complexity class P P := the class of decision problems (languages) decided by a Turing machine so that for some polynomial p and all x, the machine terminates after at most p(|x|) steps on input x. By the Polynomial Church-Turing Thesis, P is “robust” with respect to changes of the machine model. Is P also robust with respect to changes of the representation of decision problems as languages?

3 java MaxFlow 6#0|16|13|0|0|0#0|0|10|12|0|0 #0|4|0|0|14|0#0|0|9|0|0|20 #0|0|0|7|0|4|#0|0|0|0|0|0 How to encode max flow instance?

4 java MaxFlow #| | ||| #|| | || #|1111||| | #|| ||| #||| ||1111 #|||||

5 Ford-Fulkerson Ford-Fulkerson algorithm is not a polynomial time algorithm if input is encoded in binary. Ford-Fulkerson is a polynomial time algorithm if input is ecoded in unary.

6 Polynomial time computable maps f: {0,1}* ! {0,1}* is called polynomial time computable if for some polynomial p, - For all x, |f(x)| · p(|x|). - L f 2 P.

7 Polynomial time computable maps A map is polynomial time computable if and only if there is a Turing machine that on every input x accepts after at most a polynomial number of steps and leaves f(x) on its tape when terminating.

8 Good and polynomially equivalent representations A representation is good if the language of valid representations is in P. Two different representations of objects (say graphs, numbers) are called polynomially equivalent if we may translate between them using polynomial time computable maps. Ex: Adjacency matrices vs. Edge lists Ex: Binary vs. Decimal Counterexample: Binary vs. Unary

9 Robustness of Representation Given two good, polynomially equivalent representations of the instances of a decision problem, resulting in languages L 1 and L 2 we have L 1 2 P iff L 2 2 P.

10 Terminology When we say, “Problem X can be solved in polynomial time”, we mean L binary X 2 P, i.e., we assume binary representation of integers of input. If we want to say L unary X 2 P, i.e., assume unary representation of integers, we say “Problem X can be solved in pseudopolynomial time”,

11 Rigorous Formalization “Problems” Languages “Efficient Algorithms” Turing Machines, P “Search Problems” NP “Reductions” Polynomial Reductions “Universal Search Problems” NPC

12 Search Problems: NP L is in NP iff there is a language L’ in P and a polynomial p so that:

13 Intuition The y-strings are the possible solutions to the instance x. We require that solutions are not too long and that it can be checked efficiently if a given y is indeed a solution (we have a “simple” search problem)

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16 P vs. NP P is a subset of NP Is P=NP? Then any “simple” search problem has a polynomial time algorithm. This is the most famous open problem of mathematical computer science!

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18 If P=NP, mathematicians may be replaced by (much more reliable) computers: P=NP ) There is an algorithmic procedure that takes as input any formal math statement and always outputs its shortest formal proof in time polynomial in the length of the proof. This is usually regarded as evidence that P and NP are different. P vs. NP and mathematics

19 Rigorous Formalization “Problems” Languages “Efficient Algorithms” Turing Machines, P “Search Problems” NP “Reductions” Polynomial Reductions “Universal Search Problems” NPC

20 Reductions A reduction r of L 1 to L 2 is a polynomial time computable map so that 8 x: x 2 L 1 iff r(x) 2 L 2 We write L 1 · L 2 if there is a reduction of L 1 to L 2. Intuition: Efficient software for L 2 can also be used to efficiently solve L 1.

21 Example L TSP · L ILP

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23 TSP as ILP, compact formulation

24 Properties of reductions Transitivity: L 1 · L 2 Æ L 2 · L 3 ) L 1 · L 3 Follows from Polynomial Church-Turing thesis.

25 Properties of reductions Downward closure of P: L 1 · L 2 Æ L 2 2 P ) L 1 2 P. Follows from Polynomial Church-Turing thesis.

26 NP-hardness A language L is called NP-hard iff 8 L’ 2 NP: L’ · L Intuition: Software for L is strong enough to be used to solve any simple search problem. Proposition: If some NP-hard language is in P, then P=NP.

27 NPC A language L 2 NP that is NP-hard is called NP-complete. NPC := the class of NP-complete problems. Proposition: L 2 NPC ) [L 2 P iff P=NP].

28 Usefulness of NPC Languages in NPC are the least likely problems in NP to be in P. Suppose we would like to find a worst case efficient algorithm for L 2 NPC. If we believe that P is not NP, we know that no worst case efficient algorithm exists. If we have no opinion about P vs. NP, we know that if we find a worst case efficient algorithm for L, we’ll earn $1,000,000.

29 How to establish NP-hardness Thousands of natural problems are NP-complete: Empiric fact: Most natural problems in NP are either in P or NP-hard. Lemma: If L 1 is NP-hard and L 1 · L 2 then L 2 is NP-hard. We need to establish one problem to be NP-hard, the rest follows using chains of reductions. Cook (1972) established SAT to be NP-hard.

30 SAT ILP MILP MAX INDEPENDENT SET MIN VERTEX COLORING HAMILTONIAN CYCLE TSP TRIPARTITE MATCHING SET COVER KNAPSACK BINPACKING