Computational Complexity

Transcription

1 Computational Complexity

2 Algorithm performance and difficulty of problems So far we have seen problems admitting fast algorithms flow problems, shortest path, spanning tree... and other problems for which we could not do better than solving with less efficient algorithms knapsack, TSP, integer programming... Can we classify problems depending on that?

3 Easy and difficult problems Border between easy and difficult problems: existence of a polynomial time algorithm in the input size The existence of a polynomial time algorithm is easy to certify (the algorithm itself is the certificate) The non existence of a polynomial time algorithm cannot be certified Computational complexity theory [Garey and Johnson 1979]

4 A few definitions Problem: question expressed in abstract terms whose answer depends on a set of input parameters Instance of a problem: specification of the input parameters Algorithm for problem P: finite set of instructions that, if applied to any instance of P, provides in a finite number of steps a solution to the input instance or indicate that no solution exists

5 Computational models Simple models with infinite memory that can simulate any (sequential) real computer Turing machine Random Access Machine Pointer machine...

6 Complexity measure of one algorithm Number of elemental operations performed by the algorithm worst case express the number of operations in function of the input size give the asymptotic measure (neglect the constants and the lower order term) O(f(n)) notation

7 Why polynomial algorithms are better Computer performing 1M operations per second Size of the input n s s s n s s s n 5 0.1s 24.3s 13m 2 n 0.001s 17.9m 36600y 3 n 0.059s 6.5y 1.3x10 15 y Computing time

8 Why polynomial algorithms are better present computer 100 times faster 1000 times faster n N1 100 N N1 n 2 N2 10 N N2 n 5 N3 2.5 N N3 2 n N4 N4 + 6 N n N5 N5 + 4 N5 + 6 size of the largest solvable instance in 1 h

9 Pseudopolynomial algorithms Example: algorithm that solves the knapsack reducing the problem to a shortest (longest) path on an acyclic graph O(nb) complexity Polynomial in the input but not in the input size (unless unary encoding is considered)

10 Classification of the problems Problems in decision form: the expected answer is yes or no Examples connectivity problem scheduling problem SAT How to deal with optimization problems? Decision version of the problem: fix an additional parameter k. Does there exist a solution with value k? (for min problems) If we are able to solve the decision version of the problem we can solve efficiently also the optimization version by binary search over k

11 Class NP A problem P belongs to the class NP if it is in decision form it is certifiable in polynomial time Given an instance of problem P and one possible solution x There is a polynomial time algorithm that checks that solution x gives a YES answer to the given instance

12 Class P A problem P belongs to the class P if it belongs to NP there is a polynomial time algorithm that solves any instance of P P NP

13 Polynomial reduction Let P and Q NP P reduces in polynomial time to Q P Q Supposing that there exist a polynomial time (including constant) algorithm AQ solving Q then we can devise one algorithm AP that calls AQ a polynomial number of times and solves P Given an instance of P we can transform it in polynomial time into an instance of Q. The solution of Q can be transformed into a solution of P in polynomial time

14 Consequences of polynomial reductions P Q if there is a polynomial algorithm for Q P P if we can prove that P has no polynomial algorithm (hard to say) Q P is reflexive and transitive

15 Class NP-complete Q NP-complete if Q NP for each problem P NP P Q This implies that if Q could be solved in polynomial time then any problem could be solved in polynomial time thus P = NP

16 How can we prove that a problem is NP-complete? Making the reductions from all NP problems (even those which have not been conceived yet) seems impractical Exploit the transitivity of make the reduction from a known NP-complete problem SAT is NP-complete [Cook-Levin 1971]