Yossi Azar

On-line routing of virtual circuits with applications to load balancing and machine scheduling

In this paper we study the problem of on-line allocation of routes to virtual circuits (both <italic>point-to-point</italic> and <italic>multicast</italic>) where the goal is to route all requests while minimizing the required bandwidth. We concentrate on the case of <italic>Permanent</italic> virtual circuits (i.e., once a circuit is established it exists forever), and describe an algorithm that achieves on <italic>O</italic> (log <italic>n</italic>) competitive ratio with respect to maximum congestin, where <italic>n</italic>is the number of nodes in the network. Informally, our results show that instead of knowing all of the future requests, it is sufficient to increase the bandwidth of the communication links by an <italic>O</italic> (log <italic>n</italic>) factor. We also show that this result is tight, that is, for any on-line algorithm there exists a scenario in which ***(log <italic>n</italic>) increase in bandwidth is necessary in directed networks. We view virtual circuit routing as a generalization of an on-line load balancing problem, defined as follows: jobs arrive on line and each job must be assigned to one of the machines immediately upon arrival. Assigning a job to a machine increases the machines load by an amount that depends both on the job and on the machine. The goal is to minimize the maximum load. For the <italic>related machines</italic> case, we describe the first algorithm that achieves constant competitive ratio. for the <italic>unrelated</italic> case (with <italic>n</italic>machines), we describe a new method that yields <italic>O</italic>(log<italic>n</italic>)-competitive algorithm. This stands in contrast to the natural greed approach, whose competitive ratio is exactly <italic>n</italic>. show that this result is tight, that is, for any on-line algorithm there exists a scenario in which ***(log <italic>n</italic>) increase in bandwidth is necessary in directed networks.

On-line load balancing with applications to machine scheduling and virtual circuit routing

In this paper we study an idealized problem of on-line allocation of routes to virtual circuits where the goal is to minimize the required bandwidth. For the case where virtual circuits continue to exist forever, we describe an algorithm that achieves an O (log n) competitive ratio, where n is the number of nodes in the network. Informally, our results show that instead of knowing all of the future requests, it is sufficient to increase the bandwidth of the communication links by an O(log n) factor. We also show that this result is tight, i.e. for any on-line algorithm there exists a scenario in which O(log n) increase in bandwidth is necessary. We view virtual circuit routing as a generalization of an on-line scheduling problem, and hence a major part of the paper focuses on development of algorithms for non-preemptive on-line scheduling for related and unrelated machines. Specialization of routing to scheduling leads us to concentrate on scheduling in the case where jobs must be assigned immediately upon arrival; assigning a job to a machine increases this machine’s load by an amount that depends both on the job and on the machine. The goal is to minimize the maximum load. For the related machines case, we describe the first algorithm that achieves constant competitive ratio. For the unrekzted case (with n machines), we describe a new method that yields O(log n)-competitive algorithm. This stands in contrast to the natural greedy approach, which we show has only a ~(n) competitive ratio. The virtual circuit routing result follows as a generalization of the unrelated machines case.

Optimal oblivious routing in polynomial time

A recent seminal result of Racke is that for any undirected network there is an oblivious routing algorithm with a polylogarithmic competitive ratio with respect to congestion. Unfortunately, Rackes construction is not polynomial time. We give a polynomial time construction that guarantees Rackes bounds, and more generally gives the true optimal ratio for any (undirected or directed) network.

Approximation schemes for scheduling on parallel machines

We discuss scheduling problems with m identical machines and n jobs where each job has to be assigned to some machine. The goal is to optimize objective functions that solely depend on the machine completion times. As a main result, we identify some conditions on the objective function, under which the resulting scheduling problems possess a polynomial-time approximation scheme. Our result contains, generalizes, improves, simplifies, and unifies many other results in this area in a natural way.

On-Line Load Balancing of Temporary Tasks

This paper considers the nonpreemptive on-line load balancing problem where tasks havelimited durationin time. Upon arrival, each task has to be immediately assigned to one of the machines, increasing the load on this machine for the duration of the task by an amount that depends on both the machine and the task. The goal is to minimize the maximum load. Azar, Broder, and Karlin studied theunknown durationcase where the duration of a task is not known upon its arrival (On-line load balancingin“Proc. 33rd IEEE Annual Symposium on Foundations of Computer Science, 1992,” pp. 218Â?225). They focused on the special case in which for each task there is a subset of machines capable of executing it, and the increase in load due to assigning the task to one of these machines depends only on the task and not on the machine. For this case, they showed anO(n2/3)- competitive algorithm, and anÂ?(n)lower bound on the competitive ratio, wherenis the number of the machines. This paper closes the gap by giving anO(n)-competitive algorithm. In addition, trying to overcome theÂ?(n)lower bound for the case of unknown task duration, this paper initiates a study of the load balancing problem for tasks withknown duration(i.e., the duration of a task becomes known upon its arrival). For this case we show anO(lognT)-competitive algorithm, whereTis the ratio of the maximum possible duration of a task to the minimum possible duration of a task. The paper explores an alternative way to overcome theÂ?(n)bound; it considers therelated machinescase with unknown task duration. In the related machines case, a task can be executed by any machine and the increase in load depends on the speed of the machine and the weight of the task. For this case the paper gives a 20-competitive algorithm and shows a lower bound of 3Â?o(1) on the competitive ratio.

Fast convergence to nearly optimal solutions in potential games

We study the speed of convergence of decentralized dynamics to approximately optimal solutions in potential games. We consider α-Nash dynamics in which a player makes a move if the improvement in his payoff is more than an α factor of his own payoff. Despite the known polynomial convergence of α-Nash dynamics to approximate Nash equilibria in symmetric congestion games [7], it has been shown that the convergence time to approximate Nash equilibria in asymmetric congestion games is exponential [25]. In contrast to this negative result, and as the main result of this paper, we show that for asymmetric congestion games with linear and polynomial delay functions, the convergence time of α-Nash dynamics to an approximate optimal solution is polynomial in the number of players, with approximation ratio that is arbitrarily close to the price of anarchy of the game. In particular, we show this polynomial convergence under the minimal liveness assumption that each player gets at least one chance to move in every T steps. We also prove that the same polynomial convergence result does not hold for (exact) best-response dynamics, showing the α-Nash dynamics is required. We extend these results for congestion games to other potential games including weighted congestion games with linear delay functions, cut games (also called party affiliation games) and market sharing games.

Balanced allocations (extended abstract)

Suppose that we sequentially place n balls into n boxes by putting each ball into a randomly chosen box. It is well known that when we are done, the fullest box has with high probability lnn/lnlnn(1 + o(1)) balls in it. Suppose instead, that for each ball we choose two boxes at random and place the ball into the one which is less full at the time of placement. We show that with high probability, the fullest box contains only lnlnn/ln2 + O(1) balls - exponentially less than before. Furthermore, we show that a similar gap exists in the infinite process, where at each step one ball, chosen uniformly at random, is deleted, and one ball is added in the manner above. We discuss consequences of this and related theorems for dynamic resource allocation, hashing, and on-line load balancing.

A general approach to online network optimization problems

We study a wide range of online graph and network optimization problems, focusing on problems that arise in the study of connectivity and cuts in graphs. In a general online network design problem, we have a communication network known to the algorithm in advance. What is not known in advance are the bandwidth or cut demands between nodes in the network. Our results include an <i>O</i>(log <i>m</i> log <i>n</i>) competitive randomized algorithm for the online non-metric facility location and for a generalization of the problem called themulticast problem. In the non-metric facility location <i>m</i> is the number of facilities and <i>n</i> is the number of clients. The competitive ratio is nearly tight. We also present an<i>O</i>(log² <i>n</i> log <i>k</i>) competitive randomized algorithm for the on-line group Steiner problem in trees and an <i>O</i>(log³ <i>n</i> log <i>k</i>)competitive randomized algorithm for the problem in general graphs, where <i>n</i> is the number of vertices in the graph and <i>k</i> is the number of groups. Finally, we design a deterministic <i>O</i>(log³ <i>n</i> log log <i>n</i>) competitive algorithm for the online multi-cut problem. Our algorithms are based on a unified framework for designing online algorithms for problems involving connectivity and cuts. We first present a general <i>O</i>(log <i>m</i>)-deterministic algorithm for generating fractional solution that satisfies the online connectivity or cut demands, where <i>m</i> is the number of edges in the graph.

On-line generalized Steiner problem

The generalized Steiner problem (GSP) is defined as follows. We are given a graph with non-negative edge weights and a set of pairs of vertices. The algorithm has to construct minimum weight subgraph such that the two nodes of each pair are connected by a path.Off-line GSP approximation algorithms were given in Agarwal et al. (SIAM J. Comput. 24(3) (1995) 440) and Goemans and Williamson (SIAM J. Comput. 24(2) (1995) 296). We consider the on-line GSP, in which pairs of vertices arrive on-line and are needed to be connected immediately.We show that the online Min-Cost (i.e. greedy) strategy for this problem has O(log2 n) competitive ratio. The previous best algorithm was O(√nlog n) competitive (Workshop on Algorithms and Data Structures, 1993, pp. 622-633). Following this work a different (non-greedy) algorithm has been shown to achieve an O(log n) competitive ratio (Proceedings of the 29th ACM Symposium on Theory of Computing, 1997, pp. 344-353).We also consider the network connectivity leasing problem which is a generalization of the GSP. Here, edges of the graph can be either bought or leased for different costs. We provide simple randomized algorithm based on on-line generalized Steiner algorithms whose competitive ratio is within a constant factor of the best competitive algorithm for the on-line GSP.

A generic scheme for building overlay networks in adversarial scenarios

This paper presents a generic scheme for a central, yet untackled issue in overlay dynamic networks: maintaining stability over long life and against malicious adversaries. The generic scheme maintains desirable properties of the underlying structure including low diameter, and efficient routing mechanism, as well as balanced node dispersal. These desired properties are maintained in a decentralized manner without resorting to global updates or periodic stabilization protocols even against an adaptive adversary that controls the arrival and departure of nodes.