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31
Simpler and better approximation algorithms for network design
 In Proceedings of the 35th Annual ACM Symposium on Theory of Computing
, 2003
"... We give simple and easytoanalyze randomized approximation algorithms for several wellstudied NPhard network design problems. Our algorithms improve over the previously best known approximation ratios. Our main results are the following. We give a randomized 3.55approximation algorithm for the c ..."
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Cited by 77 (13 self)
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We give simple and easytoanalyze randomized approximation algorithms for several wellstudied NPhard network design problems. Our algorithms improve over the previously best known approximation ratios. Our main results are the following. We give a randomized 3.55approximation algorithm for the connected facility location problem. The algorithm requires three lines to state, one page to analyze, and improves the bestknown performance guarantee for the problem. We give a 5.55approximation algorithm for virtual private network design. Previously, constantfactor approximation algorithms were known only for special cases of this problem. We give a simple constantfactor approximation algorithm for the singlesink buyatbulk network design problem. Our performance guarantee improves over what was previously known, and is an order of magnitude improvement over previous combinatorial approximation algorithms for the problem.
Approximation via costsharing: a simple approximation algorithm for the multicommodity rentorbuy problem
 In IEEE Symposium on Foundations of Computer Science (FOCS
, 2003
"... We study the multicommodity rentorbuy problem, a type of network design problem with economies of scale. In this problem, capacity on an edge can be rented, with cost incurred on a perunit of capacity basis, or bought, which allows unlimited use after payment of a large fixed cost. Given a graph ..."
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Cited by 44 (7 self)
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We study the multicommodity rentorbuy problem, a type of network design problem with economies of scale. In this problem, capacity on an edge can be rented, with cost incurred on a perunit of capacity basis, or bought, which allows unlimited use after payment of a large fixed cost. Given a graph and a set of sourcesink pairs, we seek a minimumcost way of installing sufficient capacity on edges so that a prescribed amount of flow can be sent simultaneously from each source to the corresponding sink. The first constantfactor approximation algorithm for this problem was recently given by Kumar et al. (FOCS ’02); however, this algorithm and its analysis are both quite complicated, and its performance guarantee is extremely large. In this paper, we give a conceptually simple 12approximation algorithm for this problem. Our analysis of this algorithm makes crucial use of cost sharing, the task of allocating the cost of an object to many users of the object in a “fair ” manner. While techniques from approximation algorithms have recently yielded new progress on cost sharing problems, our work is the first to show the converse— that ideas from cost sharing can be fruitfully applied in the design and analysis of approximation algorithms. 1
The PrizeCollecting Generalized Steiner Tree Problem Via A New Approach Of PrimalDual Schema
"... In this paper we study the prizecollecting version of the Generalized Steiner Tree problem. To the best of our knowledge, there is no general combinatorial technique in approximation algorithms developed to study the prizecollecting versions of various problems. These problems are studied on a cas ..."
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Cited by 40 (13 self)
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In this paper we study the prizecollecting version of the Generalized Steiner Tree problem. To the best of our knowledge, there is no general combinatorial technique in approximation algorithms developed to study the prizecollecting versions of various problems. These problems are studied on a case by case basis by Bienstock et al. [5] by applying an LProunding technique which is not a combinatorial approach. The main contribution of this paper is to introduce a general combinatorial approach towards solving these problems through novel primaldual schema (without any need to solve an LP). We fuse the primaldual schema with Farkas lemma to obtain a combinatorial 3approximation algorithm for the PrizeCollecting Generalized Steiner Tree problem. Our work also inspires a combinatorial algorithm [12] for solving a special case of Kelly’s problem [21] of pricing edges. We also consider the kforest problem, a generalization of kMST and kSteiner tree, and we show that in spite of these problems for which there are constant factor approximation algorithms, the kforest problem is much harder to approximate. In particular, obtaining an approximation factor better than O(n 1/6−ε) for kforest requires substantially new ideas including improving the approximation factor O(n 1/3−ε) for the notorious densest ksubgraph problem. We note that kforest and prizecollecting version of Generalized Steiner Tree are closely related to each other, since the latter is the Lagrangian relaxation of the former.
Oblivious network design
 In Proceedings of the 17th Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2006
"... Consider the following network design problem: given a network G = (V, E), sourcesink pairs {si, ti} arrive and desire to send a unit of flow between themselves. The cost of the routing is this: if edge e carries a total of fe flow (from all the terminal pairs), the cost is given by ∑ e ℓ(fe), wher ..."
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Cited by 31 (8 self)
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Consider the following network design problem: given a network G = (V, E), sourcesink pairs {si, ti} arrive and desire to send a unit of flow between themselves. The cost of the routing is this: if edge e carries a total of fe flow (from all the terminal pairs), the cost is given by ∑ e ℓ(fe), where ℓ is some concave cost function; the goal is to minimize the total cost incurred. However, we want the routing to be oblivious: when terminal pair {si, ti} makes its routing decisions, it does not know the current flow on the edges of the network, nor the identity of the other pairs in the system. Moreover, it does not even know the identity of the function ℓ, merely knowing that ℓ is a concave function of the total flow on the edge. How should it (obliviously) route its one unit of flow?
On the Approximability of Some Network Design Problems
"... Consider the following classical network design problem: a set of terminals T: {t.i} wants to send traffic to a "root" r in an 'xnode graph G: (V, E). Each terminal ti sends di units of traffic, and enough bandwidth has to be allocated on the edges to permit this. However, bandwidth on an edge e ca ..."
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Cited by 25 (3 self)
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Consider the following classical network design problem: a set of terminals T: {t.i} wants to send traffic to a "root" r in an 'xnode graph G: (V, E). Each terminal ti sends di units of traffic, and enough bandwidth has to be allocated on the edges to permit this. However, bandwidth on an edge e can only be allocated in integral multiples of some base capacity ue and hence provisioning k x ue bandwidth on edge e incurs a cost of [k] times the cost of that edge. The objective is a minimumcost feasible solution. This is one of many network design problems widely studied where the bandwidth allocation being governed by side constraints: edges may only allow a subset of cables to be 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 design framework, cannot be approximated better than ~(log log n) unless NP c _ OTIME(,r~°(l°gl°gl°gn)). In particular, we show that this inapproximability threshold holds for (i) the PrioritySteiner Tree problem [7], (ii) the CostDistance problem [31], and the singlesink version of an even more fundamental problem, (iii) Fixed Charge Network Flow [33]. Our results provide a further breakthrough in the understanding of the level of complexity of network design problems. These are the first nonconstant hardness results known for all these problems.
On the Integrality Gap of a Natural Formulation of the SingleSink Buyatbulk Network Design Problem
, 2001
"... We study two versions of the single sink buyatbulk network design problem. We are given a network and a single sink, and several sources which demand a certain amount of ow to be routed to the sink. ..."
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Cited by 21 (3 self)
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We study two versions of the single sink buyatbulk network design problem. We are given a network and a single sink, and several sources which demand a certain amount of ow to be routed to the sink.
Approximating BuyatBulk and Shallowlight kSteiner trees
 In Proceedings of the 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems
, 2006
"... Abstract We study two related network design problems with two cost functions. In the buyatbulk kSteiner tree problem we are given a graph G(V, E) with a set of terminals T ` V including aparticular vertex s called the root, and an integer k < = T . There are two cost functions on theedges of G ..."
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Cited by 13 (2 self)
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Abstract We study two related network design problems with two cost functions. In the buyatbulk kSteiner tree problem we are given a graph G(V, E) with a set of terminals T ` V including aparticular vertex s called the root, and an integer k < = T . There are two cost functions on theedges of G, a buy cost b: E! R+ and a distance cost r: E! R+. The goal is to find a subtree H of G rooted at s with at least k terminals so that the cost Pe2H b(e)+Pt2Ts dist(t, s) is minimized, where dist(t, s) is the distance from t to s in H with respect to the r cost. We present an O(log4 n)approximation algorithm for the buyatbulk kSteiner tree problem. The second andclosely related one is bicriteria approximation algorithm for Shallowlight kSteiner trees. In theshallowlight kSteiner tree problem we are given a graph G with edge costs b(e) and distance costs r(e), and an integer k. Our goal is to find a minimum cost (under bcost) kSteiner tree such thatthe diameter under rcost is at most some given bound D. We develop an (O(log n), O(log3 n))approximation algorithm for a relaxed version of Shallowlight kSteiner tree where the solutionhas at least k 8 terminals. Using this we obtain an (O(log 2 n), O(log4 n))approximation algorithm
Approximating some network design problems with vertex costs
 Proc. APPROXRANDOM
, 2009
"... costs ..."
Improved Approximation Algorithms for the SingleSink BuyatBulk Network Design Problems
 In Proc. 9th Scandinavian Workshop on Algorithm Theory (SWAT
, 2003
"... Consider a given undirected graph G = (V; E) with nonnegative edge costs, a root node r 2 V , and a set D V of demands with dv representing the units of ow that demand v 2 D wishes to send to the root. We are also given K types of cables, each with a speci ed capacity and cost per unit length. ..."
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Cited by 7 (1 self)
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Consider a given undirected graph G = (V; E) with nonnegative edge costs, a root node r 2 V , and a set D V of demands with dv representing the units of ow that demand v 2 D wishes to send to the root. We are also given K types of cables, each with a speci ed capacity and cost per unit length. The singlesink buyatbulk (SSBB) problem asks for a lowcost installation of cables along the edges of G, such that the demands can simultaneously send their ows to sink/root r.