Constrained Clustering Problems and Parity Games

Clustering is a fundamental tool in data mining. It partitions points into groups (clusters) and may be used to make decisions for each point based on its group. We study several clustering objectives. We begin with studying the Euclidean k-center problem. The k-center problem is a classical combinatorial optimization problem which asks to select k centers and assign each input point in a set P to one of the centers, such that the maximum distance of any input point to its assigned center is minimized. The Euclidean k-center problem assumes that the input set P is a subset of a Euclidean space and that each location in the Euclidean space can be chosen as a center. We focus on the special case with k = 1, the smallest enclosing ball problem: given a set of points in m-dimensional Euclidean space, find the smallest sphere enclosing all the points. We combine known results about convex optimization with structural properties of the smallest enclosing ball to create a new algorithm. We show that on instances with rational coefficients our new algorithm computes the exact center of the optimal solutions and has a worst-case run time that is polynomial in the size of the input. We use the new algorithm to show that we can solve the Euclidean k-center problem in polynomial time for constant k and dimension m. The general unconstrained clustering problems are mostly very well studied. The k-center problem for example allows for elegant 2-approximation algorithms(Gonzalez 1985, Hochbaum,Shmoys 1986). However, the situation becomes significantly more difficult when constraints are added to the problem. We first look at the fair clustering. The fairness constraint is motivated by the fact that the general process of computing a clustering may harm protected (minority) classes if the clustering algorithm does not adequately represent them in desirable clusters -- especially if the data is already biased. At NIPS 2017, Chierichetti et al. proposed a model for fair clustering requiring the representation in each cluster to (approximately) preserve the global fraction of each protected class. Restricting to two protected classes, they developed both a 4-approximation algorithm for the fair k-center problem and an O(t)-approximation algorithm for the fair k-median problem, where t is a parameter for the fairness model. For multiple protected classes, the best known result is a 14-approximation algorithm for fair k-center (Rösner, Schmidt 2018). We extend and improve the known results. Firstly, we give a 5-approximation algorithm for the fair k-center problem with multiple protected classes. Secondly, we propose a relaxed fairness notion under which we can give bicriteria constant-factor approximation algorithms for the fair version of all of the classical clustering objectives (k-center, k-supplier, k-median, k-means and facility location). The latter approximation algorithms are achieved by a framework that takes an arbitrary existing unfair (integral) solution and a fair (fractional) LP solution and combines them into an essentially fair clustering with a weakly supervised rounding scheme. In this way, a fair clustering can be established belatedly, in a situation where for example the centers are already fixed. The second clustering constraint we study is privacy: Here, we are asked to only open a center when at least l points will be assigned to it. We raise the question whether a general method can be derived to turn an approximation algorithm for a clustering problem with some constraints into an approximation algorithm that additionally respects privacy. We show how to combine privacy with several other constraints and obtain approximation algorithms for the k-center problem with several combinations of constraints. In this dissertation we also study parity games, a two player game played on a directed graph. We study the case in which one of the two players controls only a small number k of nodes and the other player controls the n-k other nodes of the game. Our main result is a fixed-parameter-tractable algorithm that solves bipartite parity games in time k^{O(sqrt{k})} O(n^3), and general parity games in time (p+k)^{O(sqrt{k})} O(pnm), where p is the number of distinct priorities and m is the number of edges. For all games with k = o(n) this improves the previously fastest algorithm by Jurdziński, Paterson, and Zwick (2008). We also obtain novel kernelization results and an improved deterministic algorithm for parity games on graphs with small average node-degree

Use of the TOPSIS technique to choose the best supplier of quarry natural aggregate

Purpose. All over the world, natural substance – the most consumed after water – is the aggregate. The aim of this paper is to select the best supplier of Quarry Natural Aggregate (QNA). Methods. Selection of the best supplier of QNA is performed using the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) approach, and the method of weights based on ordinal ranking of criteria, and Lagrange multiplier. Findings. In this article, the proposed Multi-Criteria Decision Making (MCDM) approach helps the decision maker(s) to choose the best supplier of QNA amongst the considered and evaluated suppliers. Originality. During negotiation with suppliers, many are the decision makers which only attach an importance at two criteria (unit price and quality, or unit price and delivery time). Thereby, other criteria are not taken into account. Consequently, supplier selection would become not-efficient. The originality of this work is based on the multi-criteria approach to choose the best supplier of QNA. Practical implications. The efficient choice of the best supplier of QNA represents a practical and economical value for the enterprises of the civil engineering, public works, railway and hydraulic works.Мета. Обґрунтування та вибір оптимального постачальника кар’єрного щебню як природного заповнювача на основі використання багатокритеріального методу. Методика. Вибір найкращого постачальника кар’єрного природного заповнювача здійснювався за допомогою багатокритеріального методу аналізу варіантів за ступенем близькості до оптимального (TOPSIS) і методу вагових коефіцієнтів на основі порядкового ранжирування критеріїв та множника Лагранжа. Результати. Підхід, що описується в статті, заснований на багатокритеріальному прийнятті рішень і дозволяє обрати кращого постачальника природного заповнювача серед наявних та розглянутих на ринку компаній. В якості ілюстрації запропонована методологія застосована до чисельного прикладу. Це дозволило визначити вагу впливових на оцінку критеріїв, оцінку значень характеристик кожного розглянутого постачальника QNA, встановлення рейтингу розглянутих постачальників QNA і вибір альтернативи {a4} в якості кращого постачальника QNA. Наукова новизна. Вперше для вибору оптимального постачальника природного заповнювача крім факторів ціни і якості встановлено характер впливу на загальну оцінку також ряду інших факторів: вартість транспортування, транспортна відстань, час доставки, гарантійна політика й рівень відхилення. У даній роботі вперше пропонується багатокритеріальний підхід до вибору оптимального постачальника природного заповнювача кар’єра. Практична значимість. Ефективний вибір постачальника природного заповнювача кар’єра важливий з практичної та економічної точок зору для підприємств у галузі цивільного будівництва, громадських робіт, залізниці та гідротехнічних споруд.Цель. Обоснование и выбор оптимального поставщика карьерного щебня как природного заполнителя на основе использования многокритериального метода. Методика. Выбор лучшего поставщика карьерного природного заполнителя осуществлялся с помощью многокритериального метода анализа вариантов по степени близости к оптимальному (TOPSIS) и метода весовых коэффициентов на основе порядкового ранжирования критериев и множителя Лагранжа. Результаты. Подход, описываемый в статье, основан на многокритериальном принятии решений и позволяет выбрать лучшего поставщика природного заполнителя среди имеющихся и рассматриваемых на рынке компаний. В качестве иллюстрации предложенная методология применена к числовому примеру. Это позволило определить вес влияющих на оценку критериев, оценку значений характеристик каждого рассматриваемого поставщика QNA, установление рейтинга рассматриваемых поставщиков QNA и выбор альтернативы {a4} в качестве лучшего поставщика QNA. Научная новизна. Впервые для выбора оптимального поставщика природного заполнителя кроме факторов цены и качества установлен характер влияния на общую оценку также ряда других факторов: стоимость транспортирования, транспортное расстояние, время доставки, гарантийная политика и уровень отклонения. В данной работе впервые предлагается многокритериальный подход к выбору оптимального поставщика природного заполнителя карьера. Практическая значимость. Эффективный выбор поставщика природного заполнителя карьера важен с практической и экономической точек зрения для предприятий в области гражданского строительства, общественных работ, железной дороги и гидротехнических сооружений.The authors thank all the colleagues which have contributed to the realization of this research work

On the Cost of Essentially Fair Clusterings

Clustering is a fundamental tool in data mining. It partitions points into groups (clusters) and may be used to make decisions for each point based on its group. However, this process may harm protected (minority) classes if the clustering algorithm does not adequately represent them in desirable clusters -- especially if the data is already biased. At NIPS 2017, Chierichetti et al. proposed a model for fair clustering requiring the representation in each cluster to (approximately) preserve the global fraction of each protected class. Restricting to two protected classes, they developed both a 4-approximation for the fair $k$-center problem and a $O(t)$-approximation for the fair $k$-median problem, where $t$ is a parameter for the fairness model. For multiple protected classes, the best known result is a 14-approximation for fair $k$-center. We extend and improve the known results. Firstly, we give a 5-approximation for the fair $k$-center problem with multiple protected classes. Secondly, we propose a relaxed fairness notion under which we can give bicriteria constant-factor approximations for all of the classical clustering objectives $k$-center, $k$-supplier, $k$-median, $k$-means and facility location. The latter approximations are achieved by a framework that takes an arbitrary existing unfair (integral) solution and a fair (fractional) LP solution and combines them into an essentially fair clustering with a weakly supervised rounding scheme. In this way, a fair clustering can be established belatedly, in a situation where the centers are already fixed

Simulation of learning in supply partnerships

This paper introduces a general, formal treatment of dynamic constraints, i.e., constraints on the state changes that are allowed in a given state space. Such dynamic constraints can be seen as representations of "real world" constraints in a managerial context. The notions of transition, reversible and irreversible transition, and transition relation will be introduced. The link with Kripke models (for modal logics) is also made explicit. Several (subtle) examples of dynamic constraints will be given. Some important classes of dynamic constraints in a database context will be identified, e.g. various forms of cumulativity, non-decreasing values, constraints on initial and final values, life cycles, changing life cycles, and transition and constant dependencies. Several properties of these dependencies will be treated. For instance, it turns out that functional dependencies can be considered as "degenerated" transition dependencies. Also, the distinction between primary keys and alternate keys is reexamined, from a dynamic point of view.

Approximation algorithms for stochastic clustering

We consider stochastic settings for clustering, and develop provably-good approximation algorithms for a number of these notions. These algorithms yield better approximation ratios compared to the usual deterministic clustering setting. Additionally, they offer a number of advantages including clustering which is fairer and has better long-term behavior for each user. In particular, they ensure that *every user* is guaranteed to get good service (on average). We also complement some of these with impossibility results