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32
Approximation Algorithms for Data Placement in Arbitrary Networks
 in Proceedings of the 12th Annual ACMSIAM Symposium on Discrete Algorithms
, 2001
"... Abstract We develop approximation algorithms for the problem of placing replicated data in arbitrary networks, where the nodes may both issue requests for data objects and have capacity for storing data objects, so as to minimize the average dataaccess cost. We introduce the data placement problem ..."
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Abstract We develop approximation algorithms for the problem of placing replicated data in arbitrary networks, where the nodes may both issue requests for data objects and have capacity for storing data objects, so as to minimize the average dataaccess cost. We introduce the data placement problem tomodel this problem. We have a set of caches F, a set of clients D, and a set of data objects O. Each cache i can store at most ui data objects. Each client j 2 D has demand dj for a specific data object o(j) 2 O and has to be assigned to a cache that stores that object. Storing an object o in cache i incurs astorage cost of f oi, and assigning client j to cache i incurs an access cost of djcij. The goal is to find aplacement of the data objects to caches respecting the capacity constraints, and an assignment of clients
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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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.
Improved Approximation for Universal Facility Location
"... The Universal Facility Location problem (UniFL) is a generalized formulation which contains several variants of facility location including capacitated facility location (1CFL) as its special cases. We present a 6 + ~ approximation for the UniFL problem, thus improving the 8 % e approximation giv ..."
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Cited by 12 (0 self)
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The Universal Facility Location problem (UniFL) is a generalized formulation which contains several variants of facility location including capacitated facility location (1CFL) as its special cases. We present a 6 + ~ approximation for the UniFL problem, thus improving the 8 % e approximation given by iVlahdian and Pal. Our result bridges the existing gap between the UniFL problem and the 1CFL problem.
LowerBounded Facility Location
 SODA 2008
, 2008
"... We study the lowerbounded facility location problem, which generalizes the classical uncapacitated facility location problem in that it comes with lower bound constraints for the number of clients assigned to a facility in the case that this facility is opened. This problem was introduced independe ..."
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We study the lowerbounded facility location problem, which generalizes the classical uncapacitated facility location problem in that it comes with lower bound constraints for the number of clients assigned to a facility in the case that this facility is opened. This problem was introduced independently in the papers by Karger and Minkoff [12] and by Guha, Meyerson, and Munagala [7], both of which give bicriteria approximation algorithms for it. These bicriteria algorithms come within a constant factor of the optimal solution cost, but they also violate the lower bound constraints by a constant factor. Our result in this paper is the first true approximation algorithm for the lowerbounded facility location problem, which respects the lower bound constraints and achieves a constant approximation ratio for the objective function. The main technical idea for the design of the algorithm is a reduction to the capacitated facility location problem, which has known constantfactor approximation algorithms. 1
The assignment problem in content distribution networks: Unsplittable hardcapacitated facility location
 in Proc. of ACMSIAM SODA
, 2009
"... In a Content Distribution Network (CDN), there are m servers storing the data; each of them has a specific bandwidth. All the requests from a particular client should be assigned to one server, because of the routing protocol used. The goal is to minimize the total cost of these assignments —cost of ..."
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Cited by 9 (4 self)
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In a Content Distribution Network (CDN), there are m servers storing the data; each of them has a specific bandwidth. All the requests from a particular client should be assigned to one server, because of the routing protocol used. The goal is to minimize the total cost of these assignments —cost of each is proportional to the distance as well as the request size — while the load on each server is kept below its bandwidth limit. When each server also has a setup cost, this is an unsplittable hardcapacitated facility location problem. As much attention as facility location problems have received, there has been no nontrivial approximation algorithm when we have hard capacities (i.e., there can only be one copy of each facility whose capacity cannot be violated) and demands are unsplittable (i.e., all the demand from a client has to be assigned to a single facility). We observe it is NPhard to approximate the cost to within any bounded factor. Thus, for an arbitrary constant ɛ> 0, we relax the capacities to a 1 + ɛ factor. For the case where capacities are almost uniform, we give a bicriteria O(log n, 1+ɛ)approximation algorithm for general metrics and a (1 + ɛ, 1 + ɛ)approximation algorithm for tree metrics. A bicriteria (α, β)approximation algorithm produces a solution of cost at most α times the optimum, while violating the capacities by no more than a β factor. We can get the same guarantee for nonuniform capacities if we allow quasipolynomial running time. In our algorithm, some clients guess the facility they are assigned to, and facilities decide the size of the clients they serve. A straightforward approach results in exponential running time. When costs do not satisfy metricity, we show that a 1.5 violation of capacities is necessary to obtain any approximation. It is worth noting that our results generalize bin packing (zero cost matrix and facility costs equal to one), knapsack (single facility with all costs being zero), minimum makespan scheduling for related machines (all costs being zero) and some facility location problems. Key words: approximation algorithm, PTAS, network, facility location
Adding Capacity Points to a Wireless Mesh Network Using Local Search
"... Abstract — Wireless mesh network deployments are popular as a costeffective means to provide broadband connectivity to large user populations. As the network usage grows, network planners need to evolve an existing mesh network to provide additional capacity. In this paper, we study the problem of ..."
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Abstract — Wireless mesh network deployments are popular as a costeffective means to provide broadband connectivity to large user populations. As the network usage grows, network planners need to evolve an existing mesh network to provide additional capacity. In this paper, we study the problem of adding new capacity points (e.g., gateway nodes) to an existing mesh network. We first present a new technique for calculating gatewaylimited fair capacity as a function of the contention at each gateway. Then, we present two online gateway placement algorithms that use local search operations to maximize the capacity gain on an existing network. A key challenge is that each gateway’s capacity depends on the locations of other gateways and cannot be known in advance of determining a gateway placement. We address this challenge with two placement algorithms with different approaches to estimating the unknown gateway capacities. Our first placement algorithm, MinHopCount, is adapted from a solution to the facility location problem. MinHopCount minimizes path lengths and iteratively estimates the wireless capacity of each gateway location. Our second algorithm, MinContention, is adapted from a solution to the uncapacitated kmedian problem and minimizes average contention on mesh nodes, i.e. the number of links in contention range of a mesh node and the number of routes using each link. We show that our gateway placement algorithms outperform a greedy heuristic by up to 64 % on realistic topologies. For an example topology, we study the set of all possible gateway placements and find that there is large capacity gain between nearoptimal and optimal placements, but the nearoptimal placements found by local search are similar in configuration to the optimal. I.
LocalSearch based Approximation Algorithms for Mobile Facility Location Problems
"... We consider the mobile facility location (MFL) problem. We are given a set of facilities and clients located in a common metric space G = (V, c). The goal is to move each facility from its initial location to a destination (in V) and assign each client to the destination of some facility so as to mi ..."
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Cited by 4 (1 self)
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We consider the mobile facility location (MFL) problem. We are given a set of facilities and clients located in a common metric space G = (V, c). The goal is to move each facility from its initial location to a destination (in V) and assign each client to the destination of some facility so as to minimize the sum of the movementcosts of the facilities and the clientassignment costs. This abstracts facilitylocation settings where one has the flexibility of moving facilities from their current locations to other destinations so as to serve clients more efficiently by reducing their assignment costs. We give the first localsearch based approximation algorithm for this problem and achieve the bestknown approximation guarantee. Our main result is (3 + ɛ)approximation for this problem for any constant ɛ> 0 using local search. The previous best guarantee for MFL was an 8approximation algorithm due to Friggstad and Salavatipour [12] based on LProunding. Our guarantee matches the bestknown approximation guarantee for the kmedian problem. Since there is an approximationpreserving reduction from the kmedian problem to MFL, any improvement of our result would imply an analogous improvement for the kmedian problem. Furthermore, our analysis is tight (up to o(1) factors) since the tight example for the localsearch based 3approximation algorithm for kmedian can be easily adapted to show that our localsearch algorithm has a tight approximation ratio of 3. Our results extend to the weighted generalization wherein each facility i has a nonnegative weight wi and the movement cost for i is wi times the distance traveled by i. In contrast to the kmedian problem, the local search procedure that moves, at each step, a constant number of facilities (to chosen destinations) and assigns each client to the nearest destination, is known to
Improved Approximation Guarantees for LowerBounded Facility Location ⋆
"... Abstract. We consider the lowerbounded facility location (LBFL) problem, which is a generalization of uncapacitated facility location (UFL), where each open facility is required to serve a certain minimum amount of demand. The current best approximation ratio for LBFL is 448 [17]. We substantially ..."
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Abstract. We consider the lowerbounded facility location (LBFL) problem, which is a generalization of uncapacitated facility location (UFL), where each open facility is required to serve a certain minimum amount of demand. The current best approximation ratio for LBFL is 448 [17]. We substantially advance the stateoftheart for LBFL by improving its approximation ratio from 448 [17] to 82.6. Our improvement comes from a variety of ideas in algorithm design and analysis, which also yield new insights into LBFL. Our chief algorithmic novelty is to present an improved method for solving a morestructured LBFL instance obtained from I via a bicriteria approximation algorithm for LBFL, wherein all clients are aggregated at a subset F ′ of facilities, each havingatleast αM colocated clients (for some α ∈ [0,1]). The algorithm in [17] proceeds by reducing I2(α) to CFL. One of our key insights is that one can reduce the resulting LBFL instance, denoted I2(α), to a problem we introduce, called capacitydiscounted UFL (CDUFL), which is a structured special case of capacitated facility location (CFL). We give a simple localsearch algorithm for CDUFL based on add, delete, and swap moves that achieves a significantlybetter approximation ratio than the currentbest approximation ratio for CFL, which is one of the reasons behind our algorithm’s improved approximation ratio. 1
LP Rounding for kCenters with Nonuniform Hard Capacities
"... Abstract—In this paper we consider a generalization of the classical kcenter problem with capacities. Our goal is to select k centers in a graph, and assign each node to a nearby center, so that we respect the capacity constraints on centers. The objective is to minimize the maximum distance a node ..."
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Abstract—In this paper we consider a generalization of the classical kcenter problem with capacities. Our goal is to select k centers in a graph, and assign each node to a nearby center, so that we respect the capacity constraints on centers. The objective is to minimize the maximum distance a node has to travel to get to its assigned center. This problem is NPhard, even when centers have no capacity restrictions and optimal factor 2 approximation algorithms are known. With capacities, when all centers have identical capacities, a 6 approximation is known with no better lower bounds than for the infinite capacity version. While many generalizations and variations of this problem have been studied extensively, no progress was made on the capacitated version for a general capacity function. We develop the first constant factor approximation algorithm for this problem. Our algorithm uses an LP rounding approach to solve this problem, and works for the case of nonuniform hard capacities, when multiple copies of a node may not be chosen and can be extended to the case when there is a hard bound on the number of copies of a node that may be selected. Finally, for nonuniform soft capacities we present a much simpler 11approximation algorithm, which we find as one more evidence that hard capacities are much harder to deal with. Keywordsapproximation algorithms; kcenter; nonuniform capacities; hard capacities; LP rounding; I.