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.

Abstract. The design of route guidance systems faces a well-known dilemma. The approach that theoretically yields the system-optimal traffic pattern may discriminate against some users in favor of others. Proposed alternate models, however, do not directly address the system perspective and may result in inferior performance. We propose a novel model and corresponding algorithms to resolve this dilemma. We present computational results on real-world instances and compare the new approach with the well-established traffic assignment model. The essence of this study is that system-optimal routing of traffic flow with explicit integration of user constraints leads to a better performance than the user equilibrium, while simultaneously guaranteeing superior fairness compared to the pure system optimum.

Abstract. According to Wardrop's first principle, agents in a congested network choose their routes selfishly, a behavior that is captured by the Nash equilibrium of the underlying noncooperative game. A Nash equilibrium does not optimize any global criterion per se, and so there is no apparent reason why it should be close to a solution of minimal total travel time, i.e. the system optimum. In this paper, we offer extensions of recent positive results on the efficiency of Nash equilibria in traffic networks. In contrast to prior work, we present results for networks with capacities and for latency functions that are nonconvex, nondifferentiable and even discontinuous.The inclusion of upper bounds on arc flows has early been recognized as an important means to provide a more accurate description of traffic flows. In this more general model, multiple Nash equilibria may exist and an arbitrary equilibrium does not need to be nearly efficient. Nonetheless, our main result shows that the best equilibrium is as efficient as in the model without capacities. Moreover, this holds true for broader classes of travel cost functions than considered hitherto.

Abstract. According to Wardrop's first principle, agents in a congested network choose their routes selfishly, a behavior that is captured by the Nash equilibrium of the underlying noncooperative game. A Nash equilibrium does not optimize any global criterion per se, and so there is no apparent reason why it should be close to a solution of minimal total travel time, i.e. the system optimum. In this paper, we offer extensions of recent positive results on the efficiency of Nash equilibria in traffic networks. In contrast to prior work, we present results for networks with capacities and for latency functions that are nonconvex, nondifferentiable and even discontinuous.The inclusion of upper bounds on arc flows has early been recognized as an important means to provide a more accurate description of traffic flows. In this more general model, multiple Nash equilibria may exist and an arbitrary equilibrium does not need to be nearly efficient. Nonetheless, our main result shows that the best equilibrium is as efficient as in the model without capacities. Moreover, this holds true for broader classes of travel cost functions than considered hitherto.

a b s t r a c tIn machine scheduling, a set of jobs must be scheduled on a set of machines so as to minimize some global objective function, such as the makespan, which we consider in this paper. In practice, jobs are often controlled by independent, selfishly acting agents, which each select a machine for processing that minimizes the (expected) completion time. This scenario can be formalized as a game in which the players are job owners, the strategies are machines, and a player's disutility is the completion time of its jobs in the corresponding schedule. The equilibria of these games may result in larger-thanoptimal overall makespan. The price of anarchy is the ratio of the worst-case equilibrium makespan to the optimal makespan. In this paper, we design and analyze scheduling policies, or coordination mechanisms, for machines which aim to minimize the price of anarchy of the corresponding game. We study coordination mechanisms for four classes of multiprocessor machine scheduling problems and derive upper and lower bounds on the price of anarchy of these mechanisms. For several of the proposed mechanisms, we also prove that the system converges to a pure-strategy Nash equilibrium in a linear number of rounds. Finally, we note that our results are applicable to several practical problems arising in communication networks.

We consider the scheduling problem of minimizing the average weighted completion time of n jobs with release dates on a single machine. We first study two linear programming relaxations of the problem, one based on a time-indexed formulation, the other on a completiontime formulation. We show their equivalence by proving that a O(n log n) greedy algorithm leads to optimal solutions to both relaxations. The proof relies on the notion of mean busy times of jobs, a concept which enhances our understanding of these LP relaxations. Based on the greedy solution, we describe two simple randomized approximation algorithms, which are guaranteed to deliver feasible schedules with expected objective value within factors of 1.7451 and 1.6853, respectively, of the optimum. They are based on the concept of common and independent a-points, respectively. The analysis implies in particular that the worst-case relative error of the LP relaxations is at most 1.6853, and we provide instances showing that it is at least e/(e-1) 1.5819. Both algorithms may be derandomized, their deterministic versions running in O(n 2) time. The randomized algorithms also apply to the on-line setting, in which jobs arrive dynamically over time and one must decide which job to process without knowledge of jobs that will be released afterwards.

We present a short, geometric proof for the price-of-anarchy results that have recently been established in a series of papers on selfish routing in multicommodity flow networks and on nonatomic congestion games. This novel proof also facilitates two new types of theoretical results: On the one hand, we give pseudo-approximation results that depend on the class of allowable cost functions. On the other hand, we derive stronger bounds on the inefficiency of equilibria for situations in which the equilibrium costs are within reasonable limits of the fixed costs. These tighter bounds help to explain empirical observations in vehicular traffic networks. Our analysis holds in the more general context of nonatomic congestion games, which provide the framework in which we describe this work.

We consider the problem of finding near-optimal solutions for a variety of NP-hard scheduling problems for which the objective is to minimize the total weighted completion time. Recent work has led to the development of several techniques that yield constant worst-case bounds in a number of settings. We continue this line of research by providing improved performance guarantees for several of the most basic scheduling models, and by giving the first constant performance guarantee for a number of more realistically constrained scheduling problems. For example, we give an improved performance guarantee for minimizing the total weighted completion time subject to release dates on a single machine, and subject to release dates and/or precedence constraints on identical parallel machines. We also give improved bounds on the power of preemption in scheduling jobs with release dates on parallel machines.We give improved on-line algorithms for many more realistic scheduling models, including environments with parallelizable jobs, jobs contending for shared resources, tree precedence-constrained jobs, as well as shop scheduling models. In several of these cases, we give the first constant performance guarantee achieved on-line. Finally, one of the consequences of our work is the surprising structural property that there are schedules that simultaneously approximate the optimal makespan and the optimal weighted completion time to within small constants. Not only do such schedules exist, but we can find approximations to them with an on-line algorithm.