We present improved combinatorial approximation algorithms for the uncapacitated facility location and k-median problems. Two central ideas in most of our results are cost scaling and greedy improvement. We present a simple greedy local search algorithm which achieves an approximation ratio of 2:414 + in ~ O(n 2 =) time. This also yields a bicriteria approximation tradeoff of (1 +; 1+ 2=) for facility cost versus service cost which is better than previously known tradeoffs and close to the best possible. Combining greedy improvement and cost scaling with a recent primal dual algorithm for facility location due to Jain and Vazirani, we get an approximation ratio of 1.853 in ~ O(n 3 ) time. This is already very close to the approximation guarantee of the best known algorithm which is LP-based. Further, combined with the best known LP-based algorithm for facility location, we get a very slight improvement in the approximation factor for facility location, achieving 1.728....

In this paper we present a 1.52-approximation algorithm for the metric uncapacitated facility location problem, and a 2-approximation algorithm for the metric capacitated facility location problem with soft capacities. Both these algorithms improve the best previously known approximation factor for the corresponding problem, and our soft-capacitated facility location algorithm achieves the integrality gap of the standard LP relaxation of the problem. Furthermore, we will show, using a result of Thorup, that our algorithms can be implemented in quasi-linear time.

We present a simple and natural greedy algorithm for the metric uncapacitated facility location problem achieving an approximation guarantee of 1.61 whereas the best previously known was 1.73. Furthermore, we will show that our algorithm has a property which allows us to apply the technique of Lagrangian relaxation. Using this property, we can nd better approximation algorithms for many variants of the facility location problem, such as the capacitated facility location problem with soft capacities and a common generalization of the k-median and facility location problem. We will also prove a lower bound on the approximability of the k-median problem.

How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive two optimal algorithms achieving competitive ratios of 1 − 1/e for this problem. 1

Abstract We develop approximation algorithms for the problem of placing replicated data in arbitrary net-works, where the nodes may both issue requests for data objects and have capacity for storing data objects, so as to minimize the average data-access 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

We study the problem of optimally allocating online advertisement space to budget-constrained advertisers. This problem was defined and studied from the perspective of worst-case online competitive analysis by Mehta et al. Our objective is to find an algorithm that takes advantage of the given estimates of the frequencies of keywords to compute a near optimal solution when the estimates are accurate, while at the same time maintaining a good worst-case competitive ratio in case the estimates are totally incorrect. This is motivated by real-world situations where search engines have stochastic information that provide reasonably accurate estimates of the frequency of search queries except in certain highly unpredictable yet economically valuable spikes in the search pattern. Our approach is a black-box approach: we assume we have access to an oracle that uses the given estimates to recommend an advertiser every time a query arrives. We use this oracle to design an algorithm that provides two performance guarantees: the performance guarantee in the case that the oracle gives an accurate estimate, and its worst-case performance guarantee. Our algorithm can be fine tuned by adjusting a parameter α, giving a tradeoff curve between the two performance measures with the best competitive ratio for the worst-case scenario at one end of the curve and the optimal solution for the scenario where estimates are accurate at the other end. Finally, we demonstrate the applicability of our framework by applying it to two classical online problems, namely the lost cow and the ski rental problems.

We investigate variants of Lloyd’s heuristic for clustering high dimensional data in an attempt to explain its popularity (a half century after its introduction) among practitioners, and in order to suggest improvements in its application. We propose and justify a clusterability criterion for data sets. We present variants of Lloyd’s heuristic that quickly lead to provably near-optimal clustering solutions when applied to well-clusterable instances. This is the first performance guarantee for a variant of Lloyd’s heuristic. The provision of a guarantee on output quality does not come at the expense of speed: some of our algorithms are candidates for being faster in practice than currently used variants of Lloyd’s method. In addition, our other algorithms are faster on well-clusterable instances than recently proposed approximation algorithms, while maintaining similar guarantees on clustering quality. Our main algorithmic contribution is a novel probabilistic seeding process for the starting configuration of a Lloyd-type iteration. 1.

this paper, we obtain strategyproof cost allocations for two fundamental games whose underlying optimization problems are NP-hard, the set cover game and the facility location game. For the latter game, this is made possible by new approximation algorithms for the underlying optimization problem using the technique of dual fitting [7]. In retrospect, the natural greedy algorithm for the set cover problem (see [17]) can also analyzed using this technique -- we utilize this viewpoint for handling the set cover game. The facility location game was studied in [9, 4], who left the open problem of obtaining a group strategyproof mechanism based on a constant factor approximation algorithm. Our paper partially answers this question. We give a strategyproof mechanism, but cannot achieve group strategyproofness. More recently, Pal and Tardos [15] have announced a 3-approximately budget balanced crossmonotonic cost-sharing method for the facility location problem. This gives a group strategyproof mechanism for the facility location game that recovers 3 rd of the cost

In this paper we introduce a generalized version of the facility location problem in which the facility cost is a function of the number of clients assigned to the facility. We focus on the case of concave facility cost functions. We observe that this problem can be reduced to the uncapacitated facility location problem. We analyze a natural greedy algorithm for this problem and show that its approximation factor is at most 1.861. We also consider several generalizations and variants of this problem.

We apply and extend the priority algorithm framework introduced by Borodin, Nielsen and Rackoff to define "greedy-like" algorithms for (uncapacitated) facility location problems and set cover. These problems have been the focus of extensive research from the point of view of approximation algorithms, and for both problems, greedy algorithms have been proposed and analyzed. The priority algorithm definitions are general enough so as to capture a broad class of algorithms that can be characterized as "greedy-like" while still possible to derive non-trivial lower bounds on the approximability of the problems. Our results are orthogonal to complexity considerations, and hence apply to algorithms that are not necessarily polynomial-time.