Abstract. We study the load balancing problem in the context of a set of clients each wishing to run a job on a server selected among a subset of permissible servers for the particular client. We consider two different scenarios. In selfish load balancing, each client is selfish in the sense that it selects to run its job to the server among its permissible servers having the smallest latency given the assignments of the jobs of other clients to servers. In online load balancing, clients appear online and, when a client appears, it has to make an irrevocable decision and assign its job to one of its permissible servers. Here, we assume that the clients aim to optimize some global criterion but in an online fashion. A natural local optimization criterion that can be used by each client when making its decision is to assign its job to that server that gives the minimum increase of the global objective. This gives rise to greedy online solutions. The aim of this paper is to determine how much the quality of load balancing is affected by selfishness and greediness. We characterize almost completely the impact of selfishness and greediness in load balancing by presenting new and improved, tight or almost tight bounds on the price of anarchy and price of stability of selfish load balancing as well as on the competitiveness of the greedy algorithm for online load balancing when the objective is to minimize the total latency of all clients on servers with linear latency functions.

a b s t r a c tWe consider a scheduling problem in which a bounded number of jobs can be processed simultaneously by a single machine. The input is a set of n jobs J = {J 1 , . . . , J n }. Each job, J j , is associated with an interval [s j , c j ] along which it should be processed. Also given is the parallelism parameter g ≥ 1, which is the maximal number of jobs that can be processed simultaneously by a single machine. Each machine operates along a contiguous time interval, called its busy interval, which contains all the intervals corresponding to the jobs it processes. The goal is to assign the jobs to machines so that the total busy time is minimized.The problem is known to be NP-hard already for g = 2. We present a 4-approximation algorithm for general instances, and approximation algorithms with improved ratios for instances with bounded lengths, for instances where any two intervals intersect, and for instances where no interval is properly contained in another. Our study has application in optimizing the switching costs of optical networks.

Placement of regenerators in optical networks has attracted the attention of recent research works in optical networks. In this problem we are given a network, with an underlying topology of a graph G, and with a set of requests that correspond to paths in G. There is a need to put a regenerator every certain distance, because of a decrease in the power of the signal. In this work we investigate the problem of minimizing the number of locations to place the regenerators. We present analytical results regarding the complexity of this problem, in four cases, depending on whether or not there is a bound on the number of regenerators at each node, and depending on whether or not the routing is given or only the requests are given (and part of the solution is also to determine the actual routing). These results include polynomial time algorithms, NP-complete results, approximation algorithms, and inapproximability results. Copyright 2009 ACM