A Competitive Algorithm for Random-Order Stochastic Virtual Circuit Routing

We consider the virtual circuit routing problem in the stochastic model with uniformly random arrival requests. In the problem, a graph is given and requests arrive in a uniform random order. Each request is specified by its connectivity demand and the load of a request on an edge is a random variable with known distribution. The objective is to satisfy the connectivity request demands while maintaining the expected congestion (the maximum edge load) of the underlying network as small as possible. Despite a large literature on congestion minimization in the deterministic model, not much is known in the stochastic model even in the offline setting. In this paper, we present an O(log n/log log n)-competitive algorithm when optimal routing is sufficiently congested. This ratio matches to the lower bound Omega(log n/ log log n) (assuming some reasonable complexity assumption) in the offline setting. Additionally, we show that, restricting on the offline setting with deterministic loads, our algorithm yields the tight approximation ratio of Theta(log n/log log n). The algorithm is essentially greedy (without solving LP/rounding) and the simplicity makes it practically appealing

Lagrangian Duality based Algorithms in Online Energy-Efficient Scheduling

We study online scheduling problems in the general energy model of speed scaling with power down. The latter is a combination of the two extensively studied energy models, speed scaling and power down, toward a more realistic one. Due to the limits of the current techniques, only few results have been known in the general energy model in contrast to the large literature of the previous ones. In the paper, we consider a Lagrangian duality based approach to design and analyze algorithms in the general energy model. We show the applicability of the approach to problems which are unlikely to admit a convex relaxation. Specifically, we consider the problem of minimizing energy with a single machine in which jobs arrive online and have to be processed before their deadlines. We present an alpha^alpha-competitive algorithm (whose the analysis is tight up to a constant factor) where the energy power function is of typical form z^alpha + g for constants alpha > 2 and g non-negative. Besides, we also consider the problem of minimizing the weighted flow-time plus energy. We give an O(alpha/ln(alpha))-competitive algorithm; that matches (up to a constant factor) to the currently best known algorithm for this problem in the restricted model of speed scaling

Non-clairvoyant Scheduling Games

In a scheduling game, each player owns a job and chooses a machine to execute it. While the social cost is the maximal load over all machines (makespan), the cost (disutility) of each player is the completion time of its own job. In the game, players may follow selfish strategies to optimize their cost and therefore their behaviors do not necessarily lead the game to an equilibrium. Even in the case there is an equilibrium, its makespan might be much larger than the social optimum, and this inefficiency is measured by the price of anarchy -- the worst ratio between the makespan of an equilibrium and the optimum. Coordination mechanisms aim to reduce the price of anarchy by designing scheduling policies that specify how jobs assigned to a same machine are to be scheduled. Typically these policies define the schedule according to the processing times as announced by the jobs. One could wonder if there are policies that do not require this knowledge, and still provide a good price of anarchy. This would make the processing times be private information and avoid the problem of truthfulness. In this paper we study these so-called non-clairvoyant policies. In particular, we study the RANDOM policy that schedules the jobs in a random order without preemption, and the EQUI policy that schedules the jobs in parallel using time-multiplexing, assigning each job an equal fraction of CPU time

Game Efficiency Through Linear Programming Duality

The efficiency of a game is typically quantified by the price of anarchy (PoA), defined as the worst ratio of the value of an equilibrium - solution of the game - and that of an optimal outcome. Given the tremendous impact of tools from mathematical programming in the design of algorithms and the similarity of the price of anarchy and different measures such as the approximation and competitive ratios, it is intriguing to develop a duality-based method to characterize the efficiency of games. In the paper, we present an approach based on linear programming duality to study the efficiency of games. We show that the approach provides a general recipe to analyze the efficiency of games and also to derive concepts leading to improvements. The approach is particularly appropriate to bound the PoA. Specifically, in our approach the dual programs naturally lead to competitive PoA bounds that are (almost) optimal for several classes of games. The approach indeed captures the smoothness framework and also some current non-smooth techniques/concepts. We show the applicability to the wide variety of games and environments, from congestion games to Bayesian welfare, from full-information settings to incomplete-information ones

A Greedy Algorithm for Subspace Approximation Problem

In the subspace approximation problem, given m points in R^{n} and an integer k <= n, the goal is to find a k-dimension subspace of R^{n} that minimizes the l_{p}-norm of the Euclidean distances to the given points. This problem generalizes several subspace approximation problems and has applications from statistics, machine learning, signal processing to biology. Deshpande et al. [Deshpande et al., 2011] gave a randomized O(sqrt{p})-approximation and this bound is proved to be tight assuming NP != P by Guruswami et al. [Guruswami et al., 2016]. It is an intriguing question of determining the performance guarantee of deterministic algorithms for the problem. In this paper, we present a simple deterministic O(sqrt{p})-approximation algorithm with also a simple analysis. That definitely settles the status of the problem in term of approximation up to a constant factor. Besides, the simplicity of the algorithm makes it practically appealing