David B. Shmoys

A Best Possible Heuristic for the k-Center Problem

In this paper we present a 2-approximation algorithm for the k-center problem with triangle inequality. This result is “best possible” since for any δ < 2 the existence of δ-approximation algorithm would imply that P = NP. It should be noted that no δ-approximation algorithm, for any constant δ, has been reported to date. Linear programming duality theory provides interesting insight to the problem and enables us to derive, in O|E| log |E| time, a solution with value no more than twice the k-center optimal value. A by-product of the analysis is an O|E| algorithm that identifies a dominating set in G2, the square of a graph G, the size of which is no larger than the size of the minimum dominating set in the graph G. The key combinatorial object used is called a strong stable set, and we prove the NP-completeness of the corresponding decision problem.

Using dual approximation algorithms for scheduling problems theoretical and practical results

The problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well-studied problem in the theory of approximation algorithms for NP-hard optimization problems. In this paper we present the strongest possible type of result for this problem, a polynomial approximation scheme. More precisely, for each ε, we give an algorithm that runs in time O((n/ε)1/ε2) and has relative error at most ε. For algorithms that are polynomial in n and m, the strongest previously-known result was that the MULTIFIT algorithm delivers a solution with no worse than 20% relative error. In addition, we present a refinement of our scheme in the case where the performance guarantee is equal to that of MUL-TIFIT, that yields an algorithm that is both more efficient and easier to analyze than MULTIFIT. In this case, in order to guarantee a maximum relative error of 1/5+2-k, the algorithm runs in O(n(k+logn)) time. The scheme is based on a new approach to constructing approximation algorithms, which we call dual approximation algorithms, where the aim is find superoptimal, but infeasible solutions, and the performance is measured by the degree of infeasibility allowed. This notion should find wide applicability in its own right, and should be considered for any optimization problem where traditional approximation algorithms have been particularly elusive.

An approximation algorithm for the generalized assignment problem

The generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs. Each job is to be processed by exactly one machine; processing jobj on machinei requires timepij and incurs a cost ofcij; each machinei is available forTi time units, and the objective is to minimize the total cost incurred. Our main result is as follows. There is a polynomial-time algorithm that, given a valueC, either proves that no feasible schedule of costC exists, or else finds a schedule of cost at mostC where each machinei is used for at most 2Ti time units.We also extend this result to a variant of the problem where, instead of a fixed processing timepij, there is a range of possible processing times for each machine—job pair, and the cost linearly increases as the processing time decreases. We show that these results imply a polynomial-time 2-approximation algorithm to minimize a weighted sum of the cost and the makespan, i.e., the maximum job completion time. We also consider the objective of minimizing the mean job completion time. We show that there is a polynomial-time algorithm that, given valuesM andT, either proves that no schedule of mean job completion timeM and makespanT exists, or else finds a schedule of mean job completion time at mostM and makespan at most 2T.

Approximation algorithms for facility location problems

One of the most flourishing areas of research in the design and analysis of approximation algorithms has been for facility location problems. In particular, for the metric case of two simple models, the uncapacitated facility location and the k-median problems, there are now a variety of techniques that yield constant performance guarantees. These methods include LP rounding, primal-dual algorithms, and local search techniques. Furthermore, the salient ideas in these algorithms and their analyzes are simple-to-explain and reflect a surprising degree of commonality. This note is intended as companion to our lecture at CONF 2000, mainly to give pointers to the appropriate references.

Fast approximation algorithms for fractional packing and covering problems

This paper presents fast algorithms that find approximate solutions for a general class of problems, which we call fractional packing and covering problems. The only previously known algorithms for solving these problems are based on general linear programming techniques. The techniques developed in this paper greatly outperform the general methods in many applications, and are extensions of a method previously applied to find approximate solutions to multicommodity flow problems. Our algorithm is a Lagrangian relaxation technique; an important aspect of our results is that we obtain a theoretical analysis of the running time of a Lagrangian relaxation-based algorithm.We give several applications of our algorithms. The new approach yields several orders of magnitude of improvement over the best previously known running times for algorithms for the scheduling of unrelated parallel machines in both the preemptive and the nonpreemptive models, for the job shop problem, for the Held and Karp bound for the traveling salesman problem, for the cutting-stock problem, for the network embedding problem, and for the minimum-cost multicommodity flow problem.

A constant-factor approximation algorithm for the k -median problem

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 62/3-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.

Approximation algorithms for scheduling unrelated parallel machines

We consider the following scheduling problem. There are m parallel machines and n independent jobs. Each job is to be assigned to one of the machines. The processing of job j on machine i requires time pij. The objective is to find a schedule that minimizes the makespan. Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints. In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unless P = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.

Scheduling to Minimize Average Completion Time: Off-Line and On-Line Approximation Algorithms

In this paper we introduce two general techniques for the design and analysis of approximation algorithms for NP-hard scheduling problems in which the objective is to minimize the weighted sum of the job completion times. For a variety of scheduling models, these techniques yield the first algorithms that are guaranteed to find schedules that have objective function value within a constant factor of the optimum. In the first approach, we use an optimal solution to a linear programming relaxation in order to guide a simple list-scheduling rule. Consequently, we also obtain results about the strength of the relaxation. Our second approach yields on-line algorithms for these problems: in this setting, we are scheduling jobs that continually arrive to be processed and, for each time t, we must construct the schedule until time t without any knowledge of the jobs that will arrive afterwards. Our on-line technique yields constant performance guarantees for a variety of scheduling environments, and in some cases essentially matches the performance of our off-line LP-based algorithms.

Chapter 9 Sequencing and scheduling: Algorithms and complexity

Publisher Summary This chapter discusses different types of sequencing and scheduling problems, and describes different types of algorithms and the concepts of complexity theory. A class of deterministic machine scheduling problems has been introduced in the chapter. The chapter also deals with the single machine, parallel machine and multi-operation problems in this class, respectively. The two generalizations of the deterministic machine-scheduling model have been presented in the chapter. A deterministic scheduling model may give rise to various stochastic counterparts, as there is a choice in the parameters that are randomized, in their distributions, and in the classes of policies that can be applied. A characteristic feature of these models is that the stochastic parameters are regarded as independent random variables with a given distribution and that their realization occurs only after the scheduling decision has been made. In the deterministic model, one has perfect information, and capitalizing on it in minimizing the realization of a performance measure may require exponential time.

Approximation algorithms for facility location problems (extended abstract)

We present new approximation algorithms for several facility location problems. In each facility location problem that we study, there is a set of locations at which we may build a facility (such as a warehouse), where the cost of building at location i is fi; furthermore, there is a set of client locations (such as stores) that require to be serviced by a facility, and if a client at location j is assigned to a facility at location i, a cost of cij is incurred that is proportional to the distance between i and j. The objective is to determine a set of locations at which to open facilities so as to minimize the total facility and assignment costs. In the uncapacitated case, each facility can service an unlimited number of clients, whereas in the capacitated case, each facility can serve, for example, at most u clients. These models and a number of closely related ones have been studied extensively in the Operations Research literature. We shall consider the case in which the distances between locations are non-negative, symmetric and satisfy the triangle inequality. For the uncapacitated facility location, we give a polynomial-time algorithm that finds a solution of cost within a factor of 3.16 of the optimal. This is the first constant performance guarantee known for this problem. We also present approximation algorithms with constant performance guarantees for a number of capacitated models as well as a generalization in which there is a 2-level hierarchy of facilities. Our results are based on the filtering and rounding technique of Lin & Vitter. We also give a randomized variant of this technique that can then be derandomized to yield improved deterministic performance guarantees. shmoys@cs.cornell.edu. School of Operations Research & Industrial Engineering and Department of Computer Science, Cornell University, Ithaca, NY 14853. Research partially supported by NSF grants CCR-9307391 and DMS-9505155 and ONR grant N00014-96-1-0050O. yeva@cs.cornell.edu. Department of Computer Science and School of Operations Research & Industrial Engineering, Cornell University, Ithaca, NY 14853. Research partially supported by NSF grants DMI-9157199 and DMS-9505155 and ONR grant N00014-96-1-0050O. zaardal@cs.ruu.nl. Department of Computer Science, Utrecht University, Utrecht, The Netherlands. Research partially supported by NSF grant CCR-9307391, and by ESPRIT Long Term Research Project No. 20244 (project ALCOM-IT: Algorithms and Complexity in Information Technology).