Steiner Tree Approximation via Iterative Randomized Rounding

Abstract

The Steiner tree problem is one of the most fundamental NP -hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to 1.55 [Robins and Zelikovsky 2005]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP relaxation of Steiner tree with integrality gap smaller than 2 [Rajagopalan and Vazirani 1999]. In this article we present an LP-based approximation algorithm for Steiner tree with an improved approximation factor. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider an LP relaxation of the problem, which is based on the notion of directed components. We sample one component with probability proportional to the value of the associated variable in a fractional solution: the sampled component is contracted and the LP is updated consequently. We iterate this process until all terminals are connected. Our algorithm delivers a solution of cost at most ln(4) + ε < 1.39 times the cost of an optimal Steiner tree. The algorithm can be derandomized using the method of limited independence. As a by-product of our analysis, we show that the integrality gap of our LP is at most 1.55, hence answering the mentioned open question.

Covering Problems in Edge- and Node-Weighted Graphs

“…We have proven that our algorithm always finds a solution F to I and a feasible solution (ξ, ν, µ) to D(I), where both solutions satisfy (5). By the duality of LPs, the right-hand side of (5) cannot exceed the optimal value of I; thus, F is an optimal solution to I.…”

Uma breve introdução a algoritmos de aproximação

“…Existem outros algoritmos para esse problema que obtêm fatores de aproximação melhores para esse problema, sendo que o menor fator conhecidoé 1,39, de Byrka et al (2013). Para um problema de minimização, queremos um fator de aproximação tão pequeno quanto possível.…”

“…Por exemplo, obter uma 2-aproximação para o problema daárvore de Steineré tão simples quanto calcular aárvore geradora de terminais de custo mínimo (Gilbert e Pollak, 1968), enquanto obter uma 1,39-aproximação envolve estudar famílias de soluções para um problema restrito (Borchers e Du, 1997) e o desenvolvimento de técnicas avançadas de arredondamento de programa linear (Byrka et al, 2013).…”

An FPTAS for the fractional group Steiner tree problem

“…The group Steiner tree problem generalizes two important problems: The Steiner tree problem and the set cover problem. The Steiner tree problem is one of the most important NP-hard problems in combinatorial optimization that admits an approximation algorithm with a constant approximation ratio [4,5]. Actually, it is NP-hard for approximating it within a ratio of less than 96/95 [6].…”