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A ConstantFactor Approximation Algorithm for the Multicommodity RentorBuy Problem
"... ... Recent work on buyatbulk network design has concentrated on the special case where all sinks are identical; existing constantfactor approximation algorithms for this special case make crucial use of the assumption that all commodities ship flow to the same sink vertex and do not obviously ext ..."
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Cited by 100 (10 self)
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... Recent work on buyatbulk network design has concentrated on the special case where all sinks are identical; existing constantfactor approximation algorithms for this special case make crucial use of the assumption that all commodities ship flow to the same sink vertex and do not obviously extend to the multicommodity rentorbuy problem. Prior to our work, the best heuristics for the multicommodity rentorbuy problem achieved only logarithmic performance guarantees and relied on the machinery of relaxed metrical task systems or of metric embeddings. By contrast, we solve the network design problem directly via a novel primaldual algorithm.
Facility Location with Nonuniform Hard Capacities
 Proceedings of the 42nd IEEE Symposium on the Foundations of Computer Science
, 2001
"... In this paper we give the first constant factor approximation algorithm for the Facility Location Problem with nonuniform, hard capacities. Facility location problems have received a great deal of attention in recent years. Approximation algorithms have been developed for many variants. Most of thes ..."
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Cited by 54 (0 self)
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In this paper we give the first constant factor approximation algorithm for the Facility Location Problem with nonuniform, hard capacities. Facility location problems have received a great deal of attention in recent years. Approximation algorithms have been developed for many variants. Most of these algorithms are based on linear programming, but the LP techniques developed thus far have been unsuccessful in dealing with hard capacities.
Covering problems with hard capacities
 IN PROC OF. FOCS’02
, 2002
"... We consider the classical vertex cover and set cover problems with the addition of hard capacity constraints. This means that a set (vertex) can only cover a limited number of its elements (adjacent edges) and the number of available copies of each set (vertex) is bounded. This is a natural generali ..."
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Cited by 42 (1 self)
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We consider the classical vertex cover and set cover problems with the addition of hard capacity constraints. This means that a set (vertex) can only cover a limited number of its elements (adjacent edges) and the number of available copies of each set (vertex) is bounded. This is a natural generalization of the classical problems that also captures resource limitations in practical scenarios. We obtain the following results. For the unweighted vertex cover problem with hard capacities we give aapproximation algorithm which is based on randomized rounding with alterations. We prove that the weighted version is at least as hard as the set cover problem. This is an interesting separation between the approximability of weighted and unweighted versions of a “natural ” graph problem. A logarithmic approximation factor for both the set cover and the weighted vertex cover problem with hard capacities follows from the work of Wolsey [23] on submodular set cover. We provide in this paper a simple and intuitive proof for this bound.
On the Approximability of Some Network Design Problems
, 2005
"... Consider the following classical network design problem: aset of terminals T = ftig wants to send traffic to a "root" r in an nnode graph G = (V; E). Each terminal ti sends di units of traffic, and enough bandwidth has to be allocatedon the edges to permit this. However, bandwidth on an ..."
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Cited by 33 (4 self)
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Consider the following classical network design problem: aset of terminals T = ftig wants to send traffic to a "root" r in an nnode graph G = (V; E). Each terminal ti sends di units of traffic, and enough bandwidth has to be allocatedon the edges to permit this. However, bandwidth on an edge e can only be allocated in integral multiples of some basecapacity ue and hence provisioning k \Theta ue bandwidth onedge e incurs a cost of dke times the cost of that edge. Theobjective is a minimumcost feasible solution. This is one of many network design problems widelystudied where the bandwidth allocation being governed by side constraints: edges may only allow a subset of cables tobe purchased on them, or certain qualityofservice requirements may have to be met.In this work, we show that the above problem, and in fact, several basic problems in this general network designframework, cannot be approximated better than \Omega (log log n)unless NP ` DTIME \Gamma nO(log log log n) \Delta. In particular,
On Network Design Problems: Fixed Cost Flows and the Covering Steiner Problem
, 2001
"... Network design problems, such as generalizations of the Steiner Tree Problem, can be cast as edgecostow problems. An edgecost ow problem is a mincost ow problem in which the cost of the ow equals the sum of the costs of the edges carrying positive ow. ..."
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Cited by 25 (3 self)
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Network design problems, such as generalizations of the Steiner Tree Problem, can be cast as edgecostow problems. An edgecost ow problem is a mincost ow problem in which the cost of the ow equals the sum of the costs of the edges carrying positive ow.
On Allocating Goods to Maximize Fairness
, 2009
"... Given a set A of m agents and a set I of n items, where agent A ∈ A has utility uA,i for item i ∈ I, our goal is to allocate items to agents to maximize fairness. Specifically, the utility of an agent is the sum of the utilities for items it receives, and we seek to maximize the minimum utility of a ..."
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Given a set A of m agents and a set I of n items, where agent A ∈ A has utility uA,i for item i ∈ I, our goal is to allocate items to agents to maximize fairness. Specifically, the utility of an agent is the sum of the utilities for items it receives, and we seek to maximize the minimum utility of any agent. While this problem has received much attention recently, its approximability has not been wellunderstood thus far: the best known approximation algorithm achieves an Õ(√m)approximation, and in contrast, the best known hardness of approximation stands at 2. Our main result is an approximation algorithm that achieves an Õ(nɛ) approximation for any ɛ = Ω(log log n / log n) in time nO(1/ɛ). In particular, we obtain polylogarithmic approximation in quasipolynomial time, and for every constant ɛ> 0, we obtain Õ(nɛ)approximation in polynomial time. An interesting technical aspect of our algorithm is that we use as a building block a linear program whose integrality gap is Ω ( √ m). We bypass this obstacle by iteratively using the solutions produced by the LP to construct new instances with significantly smaller integrality gaps, eventually obtaining the desired approximation. We also investigate the special case of the problem, where every item has a nonzero utility for at most two agents. We show that even in this restricted setting the problem is hard to approximate upto any factor better than 2, and show a factor (2 + ɛ)approximation algorithm running in time poly(n, 1/ɛ) for any ɛ> 0. This special case can be cast as a graph edge orientation problem, and our algorithm can be viewed as a generalization of Eulerian orientations to weighted graphs. 1
LPbased approximation algorithms for capacitated facility location
 in Proc. of IPCO’04, 2004
"... In the capacitated facility location problem with hard capacities, we are given a set of facilities, F, and a set of clients D in a common metric space. Each facility i has a facility opening cost fi and capacity ui that specifies the maximum number of clients that may be assigned to this facility. ..."
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Cited by 23 (1 self)
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In the capacitated facility location problem with hard capacities, we are given a set of facilities, F, and a set of clients D in a common metric space. Each facility i has a facility opening cost fi and capacity ui that specifies the maximum number of clients that may be assigned to this facility. We want to open some facilities from the set F and assign each client to an open facility so that at most ui clients are assigned to any open facility i. The cost of assigning client j to facility i is given by the distance cij, and our goal is to minimize the sum of the facility opening costs and the client assignment costs. The only known approximation algorithms that deliver solutions within a constant factor of optimal for this NPhard problem are based on local search techniques. It is an open problem to devise an approximation algorithm for this problem based on a linear programming lower bound (or indeed, to prove a constant integrality gap for any LP relaxation). We make progress on this question by giving a 5approximation algorithm for the special case in which all of the facility costs are equal, by rounding the optimal solution to the standard LP relaxation. One notable aspect of our algorithm is that it relies on partitioning the input into a collection of singledemand capacitated facility location problems, approximately solving them, and then combining these solutions in a natural way.