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48
Simpler and better approximation algorithms for network design
 STOC'03
, 2003
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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 50 (15 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.
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 47 (8 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
Oblivious network design
"... Consider the following network design problem: given anetwork G = (V, E), sourcesink pairs {si, ti} arrive anddesire 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 allthe terminal pairs), the cost is given by P e `(fe), where ..."
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Cited by 34 (7 self)
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Consider the following network design problem: given anetwork G = (V, E), sourcesink pairs {si, ti} arrive anddesire 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 allthe terminal pairs), the cost is given by P e `(fe), where ` issome 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, itdoes 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 `, merelyknowing that ` is a concave function of the total flow on theedge. How should it (obliviously) route its one unit of flow? Can we get competitive algorithms for this problem?In this paper, we develop a framework to model oblivious network design problems (of which the above problemis a special case), and give algorithms with polylogarithmic competitive ratio for problems in this framework (and hencefor this problem). Abstractly, given a problem like the one above, the solution is a multicommodity flow producing a&quot;load &quot; on each edge of Le = `(f1(e), f2(e),..., fk(e)),and the total cost is given by an &quot;aggregation function&quot; agg(Le1,..., Lem) of the loads of all edges. Our goal is todevelop oblivious algorithms that approximately minimize the total cost of the routing, knowing the aggregation function agg, but merely knowing that ` lies in some class C, andhaving no other information about the current state of the network. Hence we want algorithms that are simultaneously&quot;functionoblivious &quot; as well as &quot;trafficoblivious&quot;. The aggregation functions we consider are the max andP objective functions, which correspond to the wellknown measures of congestion and total cost of a network; in thispaper, we prove the following: * If the aggregation function is P, we give an oblivious algorithm with
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 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 26 (4 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.
Approximation Algorithms for NodeWeighted BuyatBulk Network Design
"... We present algorithms with polylogarithmic approximation ratios for the buyatbulk network design problem in the nodeweighted setting. We obtain the following results where h is the number of pairs in the input. * An O(log h) approximation for the singlesink nonuniform buyatbulk network desig ..."
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Cited by 20 (5 self)
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We present algorithms with polylogarithmic approximation ratios for the buyatbulk network design problem in the nodeweighted setting. We obtain the following results where h is the number of pairs in the input. * An O(log h) approximation for the singlesink nonuniform buyatbulk network design. Unless P = NP this ratio is tight up to constant factors. * An O(log4 h) approximation for the multicommodity nonuniform buyatbulk network design problem.
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 o ..."
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Cited by 15 (3 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