Éva Tardos

Influential nodes in a diffusion model for social networks

We study the problem of maximizing the expected spread of an innovation or behavior within a social network, in the presence of “word-of-mouth” referral. Our work builds on the observation that individuals’ decisions to purchase a product or adopt an innovation are strongly influenced by recommendations from their friends and acquaintances. Understanding and leveraging this influence may thus lead to a much larger spread of the innovation than the traditional view of marketing to individuals in isolation. In this paper, we define a natural and general model of influence propagation that we term the decreasing cascade model, generalizing models used in the sociology and economics communities. In this model, as in related ones, a behavior spreads in a cascading fashion according to a probabilistic rule, beginning with a set of initially “active” nodes. We study the target set selection problem: we wish to choose a set of individuals to target for initial activation, such that the cascade beginning with this active set is as large as possible in expectation. We show that in the decreasing cascade model, a natural greedy algorithm is a 1-1/ e-e approximation for selecting a target set of size k.

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.

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.

Truthful mechanisms for one-parameter agents

The authors show how to design truthful (dominant strategy) mechanisms for several combinatorial problems where each agents secret data is naturally expressed by a single positive real number. The goal of the mechanisms we consider is to allocate loads placed on the agents, and an agents secret data is the cost she incurs per unit load. We give an exact characterization for the algorithms that can be used to design truthful mechanisms for such load balancing problems using appropriate side payments. We use our characterization to design polynomial time truthful mechanisms for several problems in combinatorial optimization to which the celebrated VCG mechanism does not apply. For scheduling related parallel machines (Q/spl par/C/sub max/), we give a 3-approximation mechanism based on randomized rounding of the optimal fractional solution. This problem is NP-complete, and the standard approximation algorithms (greedy load-balancing or the PTAS) cannot be used in truthful mechanisms. We show our mechanism to be frugal, in that the total payment needed is only a logarithmic factor more than the actual costs incurred by the machines, unless one machine dominates the total processing power. We also give truthful mechanisms for maximum flow, Q/spl par//spl Sigma/C/sub j/ (scheduling related machines to minimize the sum of completion times), optimizing an affine function over a fixed set, and special cases of uncapacitated facility location. In addition, for Q/spl par//spl Sigma/w/sub j/C/sub j/ (minimizing the weighted sum of completion times), we prove a lower bound of 2//spl radic/3 for the best approximation ratio achievable by truthful mechanism.

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.

Network games

Network games approach some of the traditional algorithmic questions in networks from the perspective of game theory, which gives rise of a wide range of interesting issues. In this talk we will give an overview of recent progress in many of these areas, and show strong ties to certain algorithmic techniques.

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).

Near-optimal network design with selfish agents

We introduce a simple network design game that models how independent selfish agents can build or maintain a large network. In our game every agent has a specific connectivity requirement, i.e. each agent has a set of terminals and wants to build a network in which his terminals are connected. Possible edges in the network have costs and each agents goal is to pay as little as possible. Determining whether or not a Nash equilibrium exists in this game is NP-complete. However, when the goal of each player is to connect a terminal to a common source, we prove that there is a Nash equilibrium as cheap as the optimal network, and give a polynomial time algorithm to find a (1+ε)-approximate Nash equilibrium that does not cost much more. For the general connection game we prove that there is a 3-approximate Nash equilibrium that is as cheap as the optimal network, and give an algorithm to find a (4.65+ε)-approximate Nash equilibrium that does not cost much more.

A strongly polynomial minimum cost circulation algorithm

A new algorithm is presented for the minimum cost circulation problem. The algorithm is strongly polynomial, that is, the number of arithmetic operations is polynomial in the number of nodes, and is independent of both costs and capacities.