LP-Based Approximation Algorithms for Facility Location in Buy-at-Bulk Network Design

Abstract We study problems that integrate buy-at-bulk network design into the classical (connected) facility location problem. In such problems, we need to open facilities, build a routing network, and route every client demand to an open facility. Furthermore, capacities of the edges can be purchased in discrete units from K different cable types with costs that satisfy economies of scale. We extend the linear programming frame-work of Talwar [IPCO 2002] for the single-source buy-at-bulk problem to these variants and prove integrality gap upper bounds for both facility location and connected facility location buy-at-bulk problems. For the unconnected variant we prove an integrality gap bound of O(K), and for the connected version, we get an improved bound of O(1).

On the Complexity of the Asymmetric VPN Problem

We give the first constant factor approximation algorithm for the asymmetric Virtual Private Network (VPN) problem with arbitrary concave costs. We even show the stronger result, that there is always a tree solution of cost at most 2 OPT and that a tree solution of (expected) cost at most 49.84 OPT can be determined in polynomial time. Furthermore, we answer an outstanding open question about the complexity status of the so called balanced VPN problem by proving its NP-hardness

Approximation algorithms for connected facility location with buy-at-bulk edge costs

We consider a generalization of the Connected Facility Location problem where clients may connect to open facilities via access trees shared by multiple clients. The task is to choose facilities to open, to connect these facilities by a core Steiner tree (of infinite capacity), and to design and dimension the access trees, such that the capacities installed on the edges of these trees suffice to simultaneously route all clients' demands to the open facilities. We assume that the available edge capacities are given by a set of different cable types whose costs obey economies of scale. The objective is to minimize the total cost of opening facilities, building the core Steiner tree among them, and installing capacities on the access tree edges. In this paper, we devise the first constant-factor approximation algorithm for this problem. We also present a factor 6.72 approximation algorithm for a simplified version of the problem where multiples of only one single cable type can be installed on the access edges

Network Design via Core Detouring for Problems Without a Core

Some of the currently best-known approximation algorithms for network design are based on random sampling. One of the key steps of such algorithms is connecting a set of source nodes to a random subset of them. In a recent work [Eisenbrand,Grandoni,Rothvo\ss,Schäfer-SODA'08], a new technique, \emph{core-detouring}, is described to bound the mentioned connection cost. This is achieved by defining a sub-optimal connection scheme, where paths are detoured through a proper connected subgraph (core). The cost of the detoured paths is bounded against the cost of the core and of the distances from the sources to the core. The analysis then boils down to proving the \emph{existence} of a convenient core. For some problems, such as connected facility location and single-sink rent-or-buy, the choice of the core is obvious (i.e., the Steiner tree in the optimum solution). Other, more complex network design problems do not exhibit any such core. In this paper we show that core-detouring can be nonetheless successfully applied. The basic idea is constructing a convenient core by manipulating the optimal solution in a proper (not necessarily trivial) way. We illustrate that by presenting improved approximation algorithms for two well-studied problems: virtual private network design and single-sink buy-at-bulk

Steiner Tree Approximation via Iterative Randomized Rounding

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,Zelikovsky-'05]. 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 [Vazirani,Rajagopalan-'99]. In this paper we present an LP-based approximation algorithm for Steiner tree with an improved approximation factor. Our algorithm is based on a, seemingly novel, \emph{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)+\eps<1.39 times the cost of an optimal Steiner tree. The algorithm can be derandomized using the method of limited independence. As a byproduct of our analysis, we show that the integrality gap of our LP is at most 1.55, hence answering to the mentioned open question. This might have consequences for a number of related problems