Towards an SDP-based Approach to Spectral Methods: A Nearly-Linear-Time Algorithm for Graph Partitioning and Decomposition

In this paper, we consider the following graph partitioning problem: The input is an undirected graph $G=(V,E),$ a balance parameter $b \in (0,1/2]$ and a target conductance value $\gamma \in (0,1).$ The output is a cut which, if non-empty, is of conductance at most $O(f),$ for some function $f(G, \gamma),$ and which is either balanced or well correlated with all cuts of conductance at most $\gamma.$ Spielman and Teng gave an $\tilde{O}(|E|/\gamma^{2})$-time algorithm for $f= \sqrt{\gamma \log^{3}|V|}$ and used it to decompose graphs into a collection of near-expanders. We present a new spectral algorithm for this problem which runs in time $\tilde{O}(|E|/\gamma)$ for $f=\sqrt{\gamma}.$ Our result yields the first nearly-linear time algorithm for the classic Balanced Separator problem that achieves the asymptotically optimal approximation guarantee for spectral methods. Our method has the advantage of being conceptually simple and relies on a primal-dual semidefinite-programming SDP approach. We first consider a natural SDP relaxation for the Balanced Separator problem. While it is easy to obtain from this SDP a certificate of the fact that the graph has no balanced cut of conductance less than $\gamma,$ somewhat surprisingly, we can obtain a certificate for the stronger correlation condition. This is achieved via a novel separation oracle for our SDP and by appealing to Arora and Kale's framework to bound the running time. Our result contains technical ingredients that may be of independent interest.Comment: To appear in SODA 201

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

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

Locating emergency services with priority rules: The priority queuing covering location problem

One of the assumptions of the Capacitated Facility Location Problem (CFLP) is that demand is known and fixed. Most often, this is not the case when managers take some strategic decisions such as locating facilities and assigning demand points to those facilities. In this paper we consider demand as stochastic and we model each of the facilities as an independent queue. Stochastic models of manufacturing systems and deterministic location models are put together in order to obtain a formula for the backlogging probability at a potential facility location. Several solution techniques have been proposed to solve the CFLP. One of the most recently proposed heuristics, a Reactive Greedy Adaptive Search Procedure, is implemented in order to solve the model formulated. We present some computational experiments in order to evaluate the heuristics’ performance and to illustrate the use of this new formulation for the CFLP. The paper finishes with a simple simulation exercise.Location, queuing, greedy heuristics, simulation

Privacy Preserving Clustering with Constraints

The k-center problem is a classical combinatorial optimization problem which asks to find k centers such that the maximum distance of any input point in a set P to its assigned center is minimized. The problem allows for elegant 2-approximations. However, the situation becomes significantly more difficult when constraints are added to the problem. We raise the question whether general methods can be derived to turn an approximation algorithm for a clustering problem with some constraints into an approximation algorithm that respects one constraint more. Our constraint of choice is privacy: Here, we are asked to only open a center when at least l clients will be assigned to it. We show how to combine privacy with several other constraints

Algoritmos de aproximação para problemas de localização e alocação de terminais

Orientador: Lehilton Lelis Chaves PedrosaDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: No Problema de Localização e Alocação de Terminais, a entrada é um espaço métrico composto por clientes, localidades e um conjunto de pares de clientes; uma solução é um subconjunto das localidades, onde serão abertos terminais, e uma atribuição de cada par de clientes a uma rota, que começa no primeiro cliente, passando em um ou dois terminais, e terminando no segundo cliente. O objetivo é encontrar uma solução que minimize o tamanho de todas as rotas somado com o custo de abertura de terminais. Os algoritmos de aproximação da literatura consideram apenas o caso em que o conjunto de terminais abertos é dado como parte da entrada, e o problema se torna atribuir clientes aos terminais; ou então quando o espaço é definido em classes especiais de grafos. Neste trabalho, apresentamos o primeiro algoritmo de aproximação com fator constante para o problema de, simultaneamente, escolher localidades para abrir terminais e atribuir clientes a estes. A primeira parte desta dissertação cria algoritmos de aproximação para diversas variantes do problema. A estratégia principal é reduzir os problemas de localização e alocação de terminais aos problemas clássicos de localidades, como o problema de localização de instalações e o problema das k-medianas. A redução transforma uma instância de localização e alocação de terminais em uma instância de um destes problemas, que então é resolvida usando algoritmos de aproximação já existentes na literatura. A saída do algoritmo induz uma solução para o problema original, com uma perda constante no fator de aproximação. Na segunda parte, o foco é o Problema de Localização e Alocação Única de Terminais (SAHLP), que é uma variação em que cada cliente deve estar conectado a apenas um terminal, além de não haver limite na quantidade de terminais abertos. A principal contribuição é um algoritmo 2.48-aproximado para o SAHLP, baseado em arredondamento de uma nova formulação de programa linear para o problema. O algoritmo é composto por duas fases: na primeira, a solução fracionária é escalada e um subconjunto de terminais é aberto, e na segunda, atribuímos clientes aos terminais abertos. A primeira fase segue o formato padrão de filtering para problemas de localidades. A segunda, no entanto, exigiu o desenvolvimento de novas ideias e é baseada em múltiplos critérios para realizar a atribuição. A principal técnica atribui cada cliente ao terminal aberto mais próximo, se este estiver em sua vizinhança; caso contrário, o cliente se conecta ao terminal que melhor balanceia múltiplos custos, relacionados à distância entre elesAbstract: In the Hub Location Problem (HLP), the input is a metric space composed of clients, locations and a set of pairs of clients; a solution is a subset of locations to open hubs and an assignment for each pair of clients to a route starting in the first client, passing through one or two hubs and ending in the second client. The objective is to find a solution that minimizes the length of all routes plus the cost of opening hubs. The currently known approximation algorithms consider only the case in which the set of hubs is given as part of the input and the problem is assigning clients to hubs; or when the space is defined on special classes of graphs. In this work, we present the first constant-factor approximation algorithms for the problem of, simultaneously, selecting hubs and allocating clients. The first part of the thesis derives approximation algorithms for several variants of the problem. The main strategy is to reduce the hub location problems to classical location problems, such as Facility Location and k-Median. The reduction transforms an instance of hub location into an instance of a corresponding location problem, which is then solved by known approximation algorithm. The algorithm¿s output induces a solution of the original problem within a constant loss in the approximation ratio. In the second part, we focus on the Single Allocation Hub Location Problem (SAHLP), that is the variant in which a client must be connected to only one hub and there is no limit on the number of open hubs. Our main contribution is a 2.48-approximation algorithm for the SAHLP, based on the rounding of a new linear programming formulation. The algorithm is composed of two phases: in the first one, we scale the fractional solution and open a subset of hub locations, and in the second one, we assign clients to open hubs. The first phase follows the standard filtering framework for location problems. The latter, however, demanded the development of new ideas and is based on a multiple criteria assignment. The main technique is assigning a client to a closest open hub only if there are near open hubs, and otherwise selecting the hub which balances multiple costsMestradoCiência da ComputaçãoMestre em Ciência da Computação2016/12006-1CAPESFAPES