1. Internet Engineering Czesław Smutnicki Discrete Mathematics – Computational Complexity

2. CONTENT S Asymptotic notation Decision/optimization problems Calculation models Turing machines Problem, instances, data coding Complexity classes Polynomial-time algorithms Theory of NP-completness Approximate methods Quality measures of approximation Analysis of quality measures Calculation cost Competitive analysis (on-line algorithms) Inapproximality theory

3. ASYMPTOTIC NOTATION – symbol O(n) Definition Examples

4. ASYMPTOTIC NOTATION – symbol  (n) Definition Examples

5. ASYMPTOTIC NOTATION – symbol  (n) Definition Examples

6. ASYMPTOTIC NOTATION - symbol o(n) Definition Examples

7. ASYMPTOTIC NOTATION - symbol  (n) Definition Examples

8. DECISION/OPTIMIZATION PROBLEMS decision problem: answer yes-no 2-partition problem: given numbers. Does a set exist such that optimization problem: find min or max of the goal function value knapsack problem: given numbers, and. Find the set s uch that, any optimization problem can be transformed into decision problem knapsack problem: given numbers,, and. Does a set exist s uch that,

9. CALCULATION MODELS Simple machine Finite-state machine Automata: Mealy Moore Deterministic/non-deterministic finite automata S o i io

10. DETERMINISTIC TURING MACHINE s 0 1 234 -2 …

11. NON-DETERMINISTIC TURING MACHINE s 0 1 234 -2 …

12. CODING Instance I/ Problem P Decimal coding of I Binary coding of I Unary coding of I Data string x(I) Size N(I) of the instance I Coding of numbers and structural elements

13. COMPUTATIONAL COMPLEXITY FUNCTION DEPENDS ON: Coding rule Model of calculations (DTM)

14. FUNDAMENTAL COMPLEXITY CLASSES Polynomial time algorithm O(p(n)), p – polynom ial, solvable by DTM, P class Exponential time algorithm NP class, solvable in O(p(n)) on NDTM = solvable in O(2 p(n) ) on DTM 1060 n10 -5 s6·10 -5 s n3n3 10 -3 s2·10 -1 s n5n5 10 -1 s13 m 2n2n 10 -3 s3366 y

15. NP COMPLETE PROBLEMS POLYNOMIAL TIME TRANSFORMATION PROBLEM P1 IS NP-COMPLETE IF P1 BELONGS TO NP CLASS AND FOR ANY P2 FROM NP CLASS, P2 IS POLYNOMIALLY TRANSFORMABLE TO P1 PROBLEM IS PSEUDO-POLYNOMIAL (NPI CLASS) IF ITS COMPUTATIONAL COMPLEXITY FUNCTION IS A POLYNOMIAL OF N(I) AND MAX(I)

16. COMPLEXITY CLASSES NP CLASS P CLASS NPI CLASS NP COMPLETE CLASS STRONGLY NP COMPLETE CLASS

17. Thank you for your attention DISCRETE MATHEMATICS Czesław Smutnicki