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15
Approximation algorithms for nonuniform buyatbulk network design problems
 Proc. of IEEE FOCS
"... Abstract. Buyatbulk network design problems arise in settings where the costs for purchasing or installing equipment exhibit economies of scale. The objective is to build a network of cheapest cost to support a given multicommodity flow demand between node pairs. We present approximation algorith ..."
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Cited by 58 (15 self)
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Abstract. Buyatbulk network design problems arise in settings where the costs for purchasing or installing equipment exhibit economies of scale. The objective is to build a network of cheapest cost to support a given multicommodity flow demand between node pairs. We present approximation algorithms for buyatbulk network design problems with costs on both edges and nodes of an undirected graph. Our main result is the first polylogarithmic approximation ratio for the nonuniform problem that allows different cost functions on each edge and node; the ratio we achieve is O(log4 h) where h is the number of demand pairs. In addition we present an O(log h) approximation for the single sink problem. Polylogarithmic ratios for some related problems are also obtained. Our algorithm for the multicommodity problem is obtained via a reduction to the single source problem using the notion of junction trees. We believe that this presents a simple yet useful general technique for network design problems. Key words. Nonuniform buyatbulk, network design, approximation algorithm, concave cost, network flow, economies of scale AMS subject classifications. 68Q25, 68W25, 90C27, 90C59 1. Introduction. Network
An Improved LPbased Approximation for Steiner Tree
, 2009
"... The Steiner tree problem is one of the most fundamentalhard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum weight tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from to the current best���[Robin ..."
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Cited by 28 (0 self)
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The Steiner tree problem is one of the most fundamentalhard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum weight tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from to the current best���[Robins,ZelikovskySIDMA’05]. All these algorithms are purely combinatorial. A longstanding open problem is whether there is an LPrelaxation for Steiner tree with integrality gap smaller than [Vazirani,RajagopalanSODA’99]. In this paper we improve the approximation factor for Steiner tree, developing an LPbased approximation a� algorithm. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider a directedcomponent cut relaxation for the�restricted Steiner tree problem. We sample one of these components with probability proportional to the value of the associated variable in the optimal fractional solution and contract it. We iterate this process for a proper number of times and finally output the sampled components together
Online and Stochastic Survivable Network Design
"... Consider the edgeconnectivity survivable network design problem: given a graph G = (V, E) with edgecosts, and edgeconnectivity requirements rij ∈ Z≥0 for every pair of vertices i, j ∈ V, find an (approximately) minimumcost network that provides the required connectivity. While this problem is kno ..."
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Cited by 10 (2 self)
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Consider the edgeconnectivity survivable network design problem: given a graph G = (V, E) with edgecosts, and edgeconnectivity requirements rij ∈ Z≥0 for every pair of vertices i, j ∈ V, find an (approximately) minimumcost network that provides the required connectivity. While this problem is known to admit good approximation algorithms in the offline case, no algorithms were known for this problem in the online setting. In this paper, we give a randomized O(rmax log 3 n) competitive online algorithm for this edgeconnectivity network design problem, where rmax = maxij rij. Our algorithms use the standard embeddings of graphs into random subtrees (i.e., into singly connected subgraphs) as an intermediate step to get algorithms for higher connectivity. Our results for the online problem give us approximation algorithms that admit strict costshares with the same strictness value. This, in turn, implies approximation algorithms for (a) the rentorbuy version and (b) the (twostage) stochastic version of the edgeconnected network design problem with independent arrivals. For these two problems, if we are in the case when the underlying graph is complete and the edgecosts are metric (i.e., satisfy the triangle inequality), we improve our results to give O(1)strict cost shares, which gives constantfactor rentorbuy and stochastic algorithms for these instances.
A ConstantFactor Approximation for Stochastic Steiner Forest
 STOC'09
, 2009
"... We consider the stochastic Steiner forest problem: suppose we were given a collection of Steiner forest instances, and were guaranteed that a random one of these instances would appear tomorrow; moreover, the cost of edges tomorrow will be λ times the cost of edges today. Which edges should we buy t ..."
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Cited by 2 (0 self)
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We consider the stochastic Steiner forest problem: suppose we were given a collection of Steiner forest instances, and were guaranteed that a random one of these instances would appear tomorrow; moreover, the cost of edges tomorrow will be λ times the cost of edges today. Which edges should we buy today so that we can extend it to a solution for the instance arriving tomorrow, to minimize the expected total cost? While very general results have been developed for many problems in stochastic discrete optimization over the past years, the approximation status of the stochastic Steiner Forest problem has remained open, with previous works yielding constantfactor approximations only for special cases. We resolve the status of this problem by giving a constantfactor primaldual based approximation algorithm.
Competitive Cost Sharing with Economies of Scale
"... Abstract. We consider a general class of noncooperative buyatbulk cost sharing games, in which k players must contribute to purchase a number of resources. The resources have costs and must be paid for to be available to players. Each player can specify payments and has a certain constraint on th ..."
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Cited by 1 (1 self)
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Abstract. We consider a general class of noncooperative buyatbulk cost sharing games, in which k players must contribute to purchase a number of resources. The resources have costs and must be paid for to be available to players. Each player can specify payments and has a certain constraint on the number and types of resources that she needs to have available. She strives to fulfill this constraint with the smallest investment possible. Our model includes a natural economy of scale: for a subset of players, capacity must be installed at the resources. The cost increase for larger sets of players is composed of a fixed price c(r) for each resource r and a global concave capacity function g. This cost can be shared arbitrarily between players. We consider the quality and existence of purestrategy exact and approximate Nash equilibria. In general, prices of anarchy and stability depend heavily on the economy of scale and are Θ(k/g(k)). For nonlinear functions g pure Nash equilibria might not exist and deciding their existence is NPhard. For subclasses of games corresponding to covering problems, primaldual methods can be applied to derive cheap and stable approximate Nash equilibria in polynomial time. In addition, for singleton games optimal Nash equilibria exist. In this case expensive exact as well as cheap approximate Nash equilibria can be computed in polynomial time. Some of our results can be extended to games based on facility location problems. 1
Approximability of Robust Network Design
, 2010
"... We consider robust network design problems where the set of feasible demands may be given by an arbitrary polytope or convex body more generally. This model, introduced by BenAmeur and Kerivin [2], generalizes the well studied virtual private network (VPN) problem. Most research in this area has fo ..."
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Cited by 1 (1 self)
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We consider robust network design problems where the set of feasible demands may be given by an arbitrary polytope or convex body more generally. This model, introduced by BenAmeur and Kerivin [2], generalizes the well studied virtual private network (VPN) problem. Most research in this area has focused on finding constant factor approximations for specific polytope of demands, such as the class of hose matrices used in the definition of VPN. As pointed out in [4], however, the general problem was only known to be APXhard (based on a reduction from the Steiner tree problem). We show that the general robust design is hard to approximate to within polylogarithmic factors. We establish this by showing a general reduction of buyatbulk network design to the robust network design problem. In the second part of the paper, we introduce a natural generalization of the VPN problem. In this model, the set of feasible demands is determined by a tree with edge capacities; a demand matrix is feasible if it can be routed on the tree. We give a constant factor approximation algorithm for this problem that achieves factor 8 in general, and 2 for the case where the tree has unit capacities.
God does not play dice... ALGORITHMS
, 2008
"... We consider optimization problems for which the best known approximation algorithms are randomized algorithms: these algorithms make random choices during their execution, and it has been shown that in expectation the cost of the algorithm’s solution is at most a known constant factor more than opti ..."
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We consider optimization problems for which the best known approximation algorithms are randomized algorithms: these algorithms make random choices during their execution, and it has been shown that in expectation the cost of the algorithm’s solution is at most a known constant factor more than optimal. We show how to give deterministic variants of these algorithms that have similar performance guarantees. In particular, we give conditions under which the SampleAugment algorithms proposed by Gupta et al. [42] can be derandomized, thus obtaining the best known deterministic algorithms for a number of network design problems such as the connected facility location, virtual private network design and single sink buyatbulk problems. We also give deterministic variants of the “pivoting ” algorithms proposed by Ailon et al. [4] for several ranking and clustering problems. In addition to obtaining the same performance guarantees, the analysis of our algorithms is actually simpler than that of their randomized counterparts. Finally, we take a more practical approach to one of the ranking problems considered: the rank aggregation problem. We perform an extensive evaluation of several known and new algorithms for rank aggregation on web search data. We argue that there are two important classes of algorithms for rank aggregation: positional methods and comparison sort methods. We find that hybrid algorithms, that combine a positional and comparison sort approach, work especially well on our data sets.
Deterministic Sampling Algorithms for Network Design
"... Abstract. For several NPhard network design problems, the best known approximation algorithms are remarkably simple randomized algorithms called SampleAugment algorithms in [11]. The algorithms draw a random sample from the input, solve a certain subproblem on the random sample, and augment the so ..."
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Abstract. For several NPhard network design problems, the best known approximation algorithms are remarkably simple randomized algorithms called SampleAugment algorithms in [11]. The algorithms draw a random sample from the input, solve a certain subproblem on the random sample, and augment the solution for the subproblem to a solution for the original problem. We give a general framework that allows us to derandomize most SampleAugment algorithms, i.e. to specify a specific sample for which the cost of the solution created by the SampleAugment algorithm is at most a constant factor away from optimal. Our approach allows us to give deterministic versions of the SampleAugment algorithms for the connected facility location problem, in which the open facilities need to be connected by either a tree or a tour, the virtual private network design problem, 2stage rooted stochastic Steiner tree problem with independent decisions, the a priori traveling salesman problem and the single sink buyatbulk problem. This partially answers an open question posed in Gupta et al. [11]. 1
1 An Oblivious Spanning Tree for SingleSink BuyatBulk
"... Abstract—We consider the problem of constructing a single spanning tree for the singlesink buyatbulk network design problem for doublingdimension graphs. We compute a spanning tree to route a set of demands along a graph G to or from a designated sink node. The demands could be aggregated at (or ..."
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Abstract—We consider the problem of constructing a single spanning tree for the singlesink buyatbulk network design problem for doublingdimension graphs. We compute a spanning tree to route a set of demands along a graph G to or from a designated sink node. The demands could be aggregated at (or symmetrically distributed to) intermediate edges where the fusioncost is specified by a nonnegative concave function f. We describe a novel approach for developing an oblivious spanning tree in the sense that it is independent of the number and location of data sources (or demands) and cost function at the edges. We present a deterministic, polynomialtime algorithm for constructing a spanning tree in low doublingdimension graphs that guarantees a log 3 Dapproximation over the optimal cost, where D is the diameter of the graph G. With a constant fusioncost function, our spanning tree gives a O(log 3 D)approximation for every Steiner tree that includes the sink. We also provide a Ω(logn) lowerbound for any oblivious tree in low doublingdimension graphs. To our knowledge, this is the first paper to propose a single spanning tree solution to the singlesink buyatbulk network design problem (as opposed to multiple overlay trees).