Serge A. Plotkin

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.

Set k-cover algorithms for energy efficient monitoring in wireless sensor networks

Wireless sensor networks (WSNs) are emerging as an effective means for environment monitoring. This paper investigates a strategy for energy efficient monitoring in WSNs that partitions the sensors into covers, and then activates the covers iteratively in a round-robin fashion. This approach takes advantage of the overlap created when many sensors monitor a single area. Our work builds upon previous work by Slijepcevic and Potkonjak (2001), where the model is first formulated. We have designed three approximation algorithms for a variation of the set k-cover problem, where the objective is to partition the sensors into covers such that the number of covers that include an area, summed over all areas, is maximized. The first algorithm is randomized and partitions the sensors, in expectation, within a fraction 1 - (1/e) (/spl sim/ .63) of the optimum. We present two other deterministic approximation algorithms. One is a distributed greedy algorithm with a 1/2 approximation ratio and the other is a centralized greedy algorithm with a 1 - (1/e) approximation ratio. We show that it is NP-complete to guarantee better than 15/16 of the optimal coverage, indicating that all three algorithms perform well with respect to the best approximation algorithm possible in polynomial time, assuming P /spl ne/ NP. Simulations indicate that in practice, the deterministic algorithms perform far above their worst case bounds, consistently covering more than 72% of what is covered by an optimum solution. Simulations also indicate that the increase in longevity is proportional to the amount of overlap amongst the sensors. The algorithms are fast, easy to use, and according to simulations, significantly increase the longevity of sensor networks. The randomized algorithm in particular seems quite practical.

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.

Network decomposition and locality in distributed computation

The authors introduce a concept of network decomposition, a partitioning of an arbitrary graph into small-diameter connected components, such that the graph created by contracting each component into a single node has low chromatic number. They present an efficient distributed algorithm for constructing such a decomposition and demonstrate its use for design of efficient distributed algorithms. The method yields new deterministic distributed algorithms for finding a maximal independent set in an arbitrary graph and for ( Delta +1)-coloring of graphs with maximum degree Delta . These algorithms run in O(n/sup epsilon /) time for epsilon =O((log log n/log n)/sup 1/2/), whereas the best previously known deterministic algorithms required Omega (n) time. The techniques can also be used to remove randomness from the previously known most distributed breadth-first search algorithm.<<ETX>>

Excluded minors, network decomposition, and multicommodity flow

In this paper we show that, given a graph and parameters 6 and r, we can find either a K,,. minor or an edge-cut of size O(mT/6) whose removal yields components of weak diameter O(T-26); i.e., every pair of nodes in such a component are at distance 0(r26) in the original graph. Using this lemma, we improve the best known bounds for the rein-cut max-flow ratio for mukicommodity flows in graphs with forbidden small minors. In general graphs, it was known that the ratio is O(log k) for the uniform-demand case (the case where there is a unit-demand commodity between every pair of nodes), and that the ratio is 0(log2 k) for arbitrary demands, where k is the number of commodities. In this paper we show that for graphs excluding any fixed graph as a minor (e.g. planar graphs or boundedgenus graphs), the ratio is O(1) for the uniform-demand case and O(log k) for the arbitrary demand case. For such graphs, our method yields rein-ratio cut approximation algorithms with performance bounds that match the above ratios. Computation of such cuts is a basic step for a variety of approximation algorithms for NP-complete problems.

Approximating a finite metric by a small number of tree metrics

Y. Bartal (1996, 1998) gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O (log n log log n). His result has found several applications and in particular has resulted in approximation algorithms for many graph optimization problems. However approximation algorithms based on his result are inherently randomized. In this paper we derandomize the use of Bartals algorithm in the design of approximation algorithms. We give an efficient polynomial time algorithm that given a finite n point metric G, constructs O(n log n) trees and a probability distribution /spl mu/ on them such that the expected stretch of any edge of G in a tree chosen according to /spl mu/ is at most O(log n log log n). Our result establishes that finite metrics can be probabilistically approximated by a small number of tree metrics. We obtain the first deterministic approximation algorithms for buy-at-bulk network design and vehicle routing; in addition we subsume results from our earlier work on derandomization. Our main result is obtained by a novel view of probabilistic approximation of metric spaces as a deterministic optimization problem via linear programming.

Competitive routing of virtual circuits in ATM networks

Classical routing and admission control strategies achieve provably good performance by relying on an assumption that the virtual circuits arrival pattern can be described by some a priori known probabilistic model. A new on-line routing framework, based on the notion of competitive analysis, was proposed. This framework is geared toward design of strategies that have provably good performance even in the case where there are no statistical assumptions on the arrival pattern and parameters of the virtual circuits. The on-line strategies motivated by this framework are quite different from the min-hop and reservation-based strategies. This paper surveys the on-line routing framework, the proposed routing and admission control strategies, and discusses some of the implementation issues. >

Cost-distance: two metric network design

Presents the cost-distance problem, which consists of finding a Steiner tree which optimizes the sum of edge costs along one metric and the sum of source-sink distances along an unrelated second metric. We give the first known O(log k) randomized approximation scheme for the cost-distance problem, where k is the number of sources. We reduce several common network design problems to cost-distance problems, obtaining (in some cases) the first known logarithmic approximation for them. These problems include a single-sink buy-at-bulk problem with variable pipe types between different sets of nodes, facility location with buy-at-bulk-type costs on edges, constructing single-source multicast trees with good cost and delay properties, and multi-level facility location. Our algorithm is also easier to implement and significantly faster than previously known algorithms for buy-at-bulk design problems.

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.