On the expected diameter, width, and complexity of a stochastic convex-hull

We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of $n$ points in $\mathbb{R}^d$ each of which has an existence probability, a SCH refers to the convex hull of a realization of the dataset, i.e., a random sample including each point with its existence probability. We are interested in computing certain expected statistics of a SCH, including diameter, width, and combinatorial complexity. For diameter, we establish the first deterministic 1.633-approximation algorithm with a time complexity polynomial in both $n$ and $d$. For width, two approximation algorithms are provided: a deterministic $O(1)$-approximation running in $O(n^{d+1} \log n)$ time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact $O(n^d)$-time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest

Pseudo-Separation for Assessment of Structural Vulnerability of a Network

Based upon the idea that network functionality is impaired if two nodes in a network are sufficiently separated in terms of a given metric, we introduce two combinatorial \emph{pseudocut} problems generalizing the classical min-cut and multi-cut problems. We expect the pseudocut problems will find broad relevance to the study of network reliability. We comprehensively analyze the computational complexity of the pseudocut problems and provide three approximation algorithms for these problems. Motivated by applications in communication networks with strict Quality-of-Service (QoS) requirements, we demonstrate the utility of the pseudocut problems by proposing a targeted vulnerability assessment for the structure of communication networks using QoS metrics; we perform experimental evaluations of our proposed approximation algorithms in this context

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

Training Gaussian Mixture Models at Scale via Coresets

How can we train a statistical mixture model on a massive data set? In this work we show how to construct coresets for mixtures of Gaussians. A coreset is a weighted subset of the data, which guarantees that models fitting the coreset also provide a good fit for the original data set. We show that, perhaps surprisingly, Gaussian mixtures admit coresets of size polynomial in dimension and the number of mixture components, while being independent of the data set size. Hence, one can harness computationally intensive algorithms to compute a good approximation on a significantly smaller data set. More importantly, such coresets can be efficiently constructed both in distributed and streaming settings and do not impose restrictions on the data generating process. Our results rely on a novel reduction of statistical estimation to problems in computational geometry and new combinatorial complexity results for mixtures of Gaussians. Empirical evaluation on several real-world datasets suggests that our coreset-based approach enables significant reduction in training-time with negligible approximation error

Parameterized Algorithmics for Computational Social Choice: Nine Research Challenges

Computational Social Choice is an interdisciplinary research area involving Economics, Political Science, and Social Science on the one side, and Mathematics and Computer Science (including Artificial Intelligence and Multiagent Systems) on the other side. Typical computational problems studied in this field include the vulnerability of voting procedures against attacks, or preference aggregation in multi-agent systems. Parameterized Algorithmics is a subfield of Theoretical Computer Science seeking to exploit meaningful problem-specific parameters in order to identify tractable special cases of in general computationally hard problems. In this paper, we propose nine of our favorite research challenges concerning the parameterized complexity of problems appearing in this context