A (3+ϵ)(3+\epsilon)-approximation algorithm for the minimum sum of radii problem with outliers and extensions for generalized lower bounds

For a given set of points in a metric space and an integer $k$, we seek to partition the given points into $k$ clusters. For each computed cluster, one typically defines one point as the center of the cluster. A natural objective is to minimize the sum of the cluster center's radii, where we assign the smallest radius $r$ to each center such that each point in the cluster is at a distance of at most $r$ from the center. The best-known polynomial time approximation ratio for this problem is $3.389$. In the setting with outliers, i.e., we are given an integer $m$ and allow up to $m$ points that are not in any cluster, the best-known approximation factor is $12.365$. In this paper, we improve both approximation ratios to $3+\epsilon$. Our algorithms are primal-dual algorithms that use fundamentally new ideas to compute solutions and to guarantee the claimed approximation ratios. For example, we replace the classical binary search to find the best value of a Lagrangian multiplier $\lambda$ by a primal-dual routine in which $\lambda$ is a variable that is raised. Also, we show that for each connected component due to almost tight dual constraints, we can find one single cluster that covers all its points and we bound its cost via a new primal-dual analysis. We remark that our approximation factor of $3+\epsilon$ is a natural limit for the known approaches in the literature. Then, we extend our results to the setting of lower bounds. There are algorithms known for the case that for each point $i$ there is a lower bound $L_{i}$, stating that we need to assign at least $L_{i}$ clients to $i$ if $i$ is a cluster center. For this setting, there is a $3.83$ approximation if outliers are not allowed and a ${12.365}$-approximation with outliers. We improve both ratios to $3.5 + \epsilon$ and, at the same time, generalize the type of allowed lower bounds

Um algoritmo de aproximação para o problema da árvore de Steiner q-métrico com peso nos vértices

Orientadores: Lehilton Lelis Chaves Pedrosa, Flávio Keidi MiyazawaDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O Problema da Árvore de Steiner de Custo Mínimo é um problema de otimização clássico em que, dado um grafo e subconjunto de vértices, chamados terminais, procura-se obter um subgrafo conexo com todos os terminais de forma que a soma dos pesos das arestas seja mínima. Consideramos a variante em que o grafo é completo e a função de peso sobre as arestas é uma métrica e, além disso, existe uma função de pesos não negativos sobre os vértices. O objetivo é encontrar uma árvore que contém todos os terminais e que minimiza o peso total de vértices e arestas. Nesta dissertação, observamos que o problema geral admite uma 2-aproximação simples. Para o caso particular em que o peso de um vértice é no máximo q vezes o peso da aresta mais leve, para uma constante q, obtemos um algoritmo aleatorizado baseado em arredondamento de PL com fator de aproximação 1,62 e um algoritmo guloso com fator de aproximação 1,55Abstract: The Minimum Cost Steiner Tree is a classical optimization problem where, given a graph and a subset of vertices called terminals, one is asked to find a connected subgraph spanning the set of terminals and whose edge weight is minimum. We consider the variant where the graph is complete and the edge weight function is a metric and there is a non-negative weight function on the vertices. The objective is to find a tree spanning the terminals such that the sum of edge and vertex weights is minimum. In this thesis, we observe that the general problem admits a simple 2-approximation. For the special case where the weight of a vertex is at most q times the weight of lightest edge, for a constant q, we obtain a randomized LP-rounding algorithm with approximation factor 1.62 and a greedy algorithm with approximation factor 1.55MestradoCiência da ComputaçãoMestre em Ciência da Computação162305/2015-0CNP