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νλ₯ μ΅λν μ‘°ν©μ΅μ ν λ¬Έμ μ λν κ·Όμ¬ν΄λ²
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Όλ¬Έ(μμ¬)--μμΈλνκ΅ λνμ :곡과λν μ°μ
곡νκ³Ό,2019. 8. μ΄κ²½μ.In this thesis, we consider a variant of the deterministic combinatorial optimization problem (DCO) where there is uncertainty in the data, the probability maximizing combinatorial optimization problem (PCO). PCO is the problem of maximizing the probability of satisfying the capacity constraint, while guaranteeing the total profit of the selected subset is at least a given value. PCO is closely related to the chance-constrained combinatorial optimization problem (CCO), which is of the form that the objective function and the constraint function of PCO is switched. It search for a subset that maximizes the total profit while guaranteeing the probability of satisfying the capacity constraint is at least a given threshold. Thus, we discuss the relation between the two problems and analyse the complexities of the problems in special cases. In addition, we generate pseudo polynomial time exact algorithms of PCO and CCO that use an exact algorithm of a deterministic constrained combinatorial optimization problem. Further, we propose an approximation scheme of PCO that is fully polynomial time approximation scheme (FPTAS) in some special cases that are NP-hard. An approximation scheme of CCO is also presented which was derived in the process of generating the approximation scheme of PCO.λ³Έ λ
Όλ¬Έμμλ μΌλ°μ μΈ μ‘°ν© μ΅μ ν λ¬Έμ (deterministic combinatorial optimization problem : DCO)μμ λ°μ΄ν°μ λΆνμ€μ±μ΄ μ‘΄μ¬ν λλ₯Ό λ€λ£¨λ λ¬Έμ λ‘, μ΄ μμ΅μ μ£Όμ΄μ§ μμ μ΄μμΌλ‘ 보μ₯νλ©΄μ μ©λ μ μ½μ λ§μ‘±μν¬ νλ₯ μ μ΅λννλ νλ₯ μ΅λν μ‘°ν© μ΅μ ν λ¬Έμ (probability maximizing combinatorial optimization problem : PCO)μ λ€λ£¬λ€. PCOμ λ§€μ° λ°μ ν κ΄κ³κ° μλ λ¬Έμ λ‘, μ΄ μμ΅μ μ΅λννλ©΄μ μ©λ μ μ½μ λ§μ‘±μν¬ νλ₯ μ΄ μΌμ κ° μ΄μμ΄ λλλ‘ λ³΄μ₯νλ νλ₯ μ μ½ μ‘°ν© μ΅μ ν λ¬Έμ (chance-constrained combinatorial optimization problem : CCO)κ° μλ€. μ°λ¦¬λ λ λ¬Έμ μ κ΄κ³μ λνμ¬ λ
Όμνκ³ νΉμ 쑰건 νμμ λ λ¬Έμ μ 볡μ‘λλ₯Ό λΆμνμλ€. λν, μ μ½μμ΄ νλ μΆκ°λ DCOλ₯Ό λ°λ³΅μ μΌλ‘ νμ΄ PCOμ CCOμ μ΅μ ν΄λ₯Ό ꡬνλ μ μ¬ λ€νμκ° μκ³ λ¦¬μ¦μ μ μνμλ€. λ λμκ°, PCOκ° NP-hardμΈ νΉλ³ν μΈμ€ν΄μ€λ€μ λν΄μ μμ λ€νμκ° κ·Όμ¬ν΄λ²(FPTAS)κ° λλ κ·Όμ¬ν΄λ²μ μ μνμλ€. μ΄ κ·Όμ¬ν΄λ²μ μ λνλ κ³Όμ μμ CCOμ κ·Όμ¬ν΄λ² λν κ³ μνμλ€.Chapter 1 Introduction 1
1.1 Problem Description 1
1.2 Literature Review 7
1.3 Research Motivation and Contribution 12
1.4 Organization of the Thesis 13
Chapter 2 Computational Complexity of Probability Maximizing Combinatorial Optimization Problem 15
2.1 Complexity of General Case of PCO and CCO 18
2.2 Complexity of CCO in Special Cases 19
2.3 Complexity of PCO in Special Cases 27
Chapter 3 Exact Algorithms 33
3.1 Exact Algorithm of PCO 34
3.2 Exact Algorithm of CCO 38
Chapter 4 Approximation Scheme for Probability Maximizing Combinatorial Optimization Problem 43
4.1 Bisection Procedure of rho 46
4.2 Approximation Scheme of CCO 51
4.3 Variation of the Bisection Procedure of rho 64
4.4 Comparison to the Approximation Scheme of Nikolova 73
Chapter 5 Conclusion 77
5.1 Concluding Remarks 77
5.2 Future Works 79
Bibliography 81
κ΅λ¬Έμ΄λ‘ 87Maste