Abstract not found

We present the first constant-factor approximation algorithm for the metric k-median problem. The k-median problem is one of the most well-studied clustering problems, i.e., those problems in which the aim is to partition a given set of points into clusters so that the points within a cluster are relatively close with respect to some measure. For the metric k-median problem, we are given n points in a metric space. We select k of these to be cluster centers, and then assign each point to its closest selected center. If point j is assigned to a center i, the cost incurred is proportional to the distance between i and j. The goal is to select the k centers that minimize the sum of the assignment costs. We give a 6 2 3-approximation algorithm for this problem. This improves upon the best previously known result of O(log k log log k), which was obtained by refining and derandomizing a randomized O(log n log log n)-approximation algorithm of Bartal. 1

Time-indexed linear programming formulations have recently received a great deal of attention for their practical effectiveness in solving a number of single-machine scheduling problems. We show that these formulations are also an important tool in the design of approximation algorithms with good worst-case performance guarantees. We give simple new rounding techniques to convert an optimal fractional solution into a feasible schedule for which we can prove a constant-factor performance guarantee, thereby giving the first theoretical evidence of the strength of these relaxations. Specifically, we consider the problem of minimizing the total weighted job completion time on a single machine subject to precedence constraints, and give a polynomialtime (4 + ffl)-approximation algorithm, for any ffl ? 0; the best previously known guarantee for this problem was superlogarithmic. With somewhat larger constants, we also show how to extend this result to the case with release date constraints, ...

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 study approximation algorithms for several NP-hard facility location problems. We prove that a simple local search heuristic yields polynomial-time constant-factor approximation bounds for the metric versions of the uncapacitated k-median problem and the uncapacitated facility location problem. (For the k-median problem, our algorithms require a constant-factor blowup in the parameter k.) This local search heuristic was rst proposed several decades ago, and has been shown to exhibit good practical performance in empirical studies. We also extend the above results to obtain constant-factor approximation bounds for the metric versions of capacitated k-median and facility location problems.

Recent studies on operational wireless LANs (WLANs) have shown that the traffic load is often unevenly distributed among the access points (APs). Such load imbalance results in unfair bandwidth allocation among users. We argue that the load imbalance and consequent unfair bandwidth allocation can be greatly alleviated by intelligently associating users to APs, termed association control, rather than having users associate with the APs of strongest signal strength. In this paper, we present an efficient algorithmic solution to determine the user-AP associations for max-min fair bandwidth allocation. We provide a rigorous formulation of the association control problem, considering bandwidth constraints of both the wireless and backhaul links. We show the strong correlation between fairness and load balancing, which enables us to use load balancing techniques for obtaining optimal max-min fair bandwidth allocation. As this problem is NP-hard, we devise algorithms that achieve constant-factor approximation. In particular, we present a 2-approximation algorithm for unweighted users and a 3-approximation algorithm for weighted users. In our algorithms, we first compute a fractional association solution, in which users can be associated with multiple APs simultaneously. This solution guarantees the fairest bandwidth allocation in terms of max-min fairness. Then, by utilizing a rounding method, we obtain the integral solution from the fractional solution. We also consider time fairness and present a polynomialtime algorithm for optimal integral solution. We further extend our schemes for the on-line case where users may join and leave dynamically. Our simulations demonstrate that the proposed algorithms achieve close to optimal load balancing (i.e., maxmin fairness) and they outperform commonly-used heuristic approaches.

We study problems in multiobjective optimization, in which solutions to a combinatorial optimization problem are evaluated with respect to several cost criteria, and we are interested in the trade-off between these objectives (the so-called Pareto curve). We point out that, under very general conditions, there is a polynomially succinct curve that -approximates the Pareto curve, for any > 0. We give a necessary and sucient condition under which this approximate Pareto curve can be constructed in time polynomial in the size of the instance and 1=. In the case of multiple linear objectives, we distinguish between two cases: When the underlying feasible region is convex, then we show that approximating the multi-objective problem is equivalent to approximating the single-objective problem. If, however, the feasible region is discrete, then we point out that the question reduces to an old and recurrent one: How does the complexity of a combinatorial optimization problem change when its feasible region is intresected with a hyperplane with small coefficients; we report some interesting new ndings in this domain. Finally, we apply these concepts and techniques to formulate and solve approximately a cost-time-quality trade-off for optimizing access to the world-wide web, in a model first studied by Etzioni et al [EHJ+] (which was actually the original motivation for this work).

The Multiple Knapsackproblem (MKP) is a natural and well known generalization of the single knapsack problem and is defined as follows. We are given a set of n items and m bins (knapsacks) such that each item i has a profit p(i) and a size s(i), and each bin j has a capacity c(j). The goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins. MKP is a special case of the Generalized Assignment problem (GAP) where the profit and the size of an item can vary based on the specific bin that it is assigned to. GAP is APX-hard and a 2-approximation for it is implicit in the work of Shmoys and Tardos [26], and thus far, this was also the best known approximation for MKP. The main result of this paper is a polynomial time approximation scheme for MKP. Apart from its inherent theoretical interest as a common generalization of the well-studied knapsack and bin packing problems, it appears to be the strongest special case of GAP that is not APX-hard. We substantiate this by showing that slight generalizations of MKP that are very restricted versions of GAP are APX-hard. Thus our results help demarcate the boundary at which instances of GAP becomeAPX-hard. An interesting and novel aspect of our approach is an approximation preserving reduction from an arbitrary instance of MKP to an instance with O(log n) distinct sizes and profits.

We study approximation algorithms for placing replicated data in arbitrary networks. Consider a network of nodes with individual storage capacities and a metric communication cost function, in which each node periodically issues a request for an object drawn from a collection of uniform-length objects. We consider the problem of placing copies of the objects among the nodes such that the average access cost is minimized. Our main result is a polynomial-time constant-factor approximation algorithm for this placement problem. Our algorithm is based on a careful rounding of a linear programming relaxation of the problem. We also show that the data placement problem is MAXSNP-hard. We extend our approximation result to a generalization of the data placement problem that models additional costs such as the cost of realizing the placement. We also show that when object lengths are non-uniform, a constant-factor approximation is achievable if the capacity at each node in the approximate solution is allowed to exceed that in the optimal solution by the length of the largest object.

We present a randomized procedure for rounding fractional perfect matchings to (integral) matchings. If the original fractional matching satis es any linear inequality, then with high probability, the new matching satis es that linear inequality in an approximate sense. This extends the well-known LP rounding procedure of Raghavan and Thompson, which is usually used to round fractional solutions of linear programs.