A Relaxed FPTAS for Chance-Constrained Knapsack

The stochastic knapsack problem is a stochastic version of the well known deterministic knapsack problem, in which some of the input values are random variables. There are several variants of the stochastic problem. In this paper we concentrate on the chance-constrained variant, where item values are deterministic and item sizes are stochastic. The goal is to find a maximum value allocation subject to the constraint that the overflow probability is at most a given value. Previous work showed a PTAS for the problem for various distributions (Poisson, Exponential, Bernoulli and Normal). Some strictly respect the constraint and some relax the constraint by a factor of (1+epsilon). All algorithms use Omega(n^{1/epsilon}) time. A very recent work showed a "almost FPTAS" algorithm for Bernoulli distributions with O(poly(n) * quasipoly(1/epsilon)) time. In this paper we present a FPTAS for normal distributions with a solution that satisfies the chance constraint in a relaxed sense. The normal distribution is particularly important, because by the Berry-Esseen theorem, an algorithm solving the normal distribution also solves, under mild conditions, arbitrary independent distributions. To the best of our knowledge, this is the first (relaxed or non-relaxed) FPTAS for the problem. In fact, our algorithm runs in poly(n/epsilon) time. We achieve the FPTAS by a delicate combination of previous techniques plus a new alternative solution to the non-heavy elements that is based on a non-convex program with a simple structure and an O(n^2 log {n/epsilon}) running time. We believe this part is also interesting on its own right

확률최대화 조합최적화 문제에 대한 근사해법

학위논문(석사)--서울대학교 대학원 :공과대학 산업공학과,2019. 8. 이경식.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.본 논문에서는 일반적인 조합 최적화 문제(deterministic combinatorial optimization problem : DCO)에서 데이터의 불확실성이 존재할 때를 다루는 문제로, 총 수익을 주어진 상수 이상으로 보장하면서 용량 제약을 만족시킬 확률을 최대화하는 확률 최대화 조합 최적화 문제(probability maximizing combinatorial optimization problem : PCO)을 다룬다. PCO와 매우 밀접한 관계가 있는 문제로, 총 수익을 최대화하면서 용량 제약을 만족시킬 확률이 일정 값 이상이 되도록 보장하는 확률 제약 조합 최적화 문제(chance-constrained combinatorial optimization problem : CCO)가 있다. 우리는 두 문제의 관계에 대하여 논의하고 특정 조건 하에서 두 문제의 복잡도를 분석하였다. 또한, 제약식이 하나 추가된 DCO를 반복적으로 풀어 PCO와 CCO의 최적해를 구하는 유사 다항시간 알고리즘을 제안하였다. 더 나아가, PCO가 NP-hard인 특별한 인스턴스들에 대해서 완전 다항시간 근사해법(FPTAS)가 되는 근사해법을 제안하였다. 이 근사해법을 유도하는 과정에서 CCO의 근사해법 또한 고안하였다.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 국문초록 87Maste