1. Opracowanie językowe dr inż. J. Jarnicki
Internet Engineering
Czesław Smutnicki
Discrete Mathematics – Computational Complexity

2. CONTENTS
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 such that ,
any optimization problem can be transformed into decision problem
knapsack problem: given numbers , , and .
Does a set exist such that ,

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

10. DETERMINISTIC TURING MACHINE-2 -1 1 2 3 4 …

11. NON-DETERMINISTIC TURING MACHINE-2 -1 1 2 3 4 …

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 CLASSES10 60 n
10-5 s
6·10-5 s n3
10-3 s
2·10-1 s n5
10-1 s
13 m 2n
3366 y
Polynomial time algorithm O(p(n)), p – polynomial, solvable by DTM, P class
Exponential time algorithm
NP class, solvable in O(p(n)) on NDTM = solvable in O(2p(n)) on DTM

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 NPI CLASS NP COMPLETE CLASS P CLASS
STRONGLY NP COMPLETE CLASS

17. Thank you for your attention
DISCRETE MATHEMATICS
Czesław Smutnicki