Alexander Souza

The bell is ringing in speed-scaled multiprocessor scheduling

This paper investigates the problem of scheduling jobs on multiple speed-scaled processors without migration, i.e., we have constant α > 1 such that running a processor at speed <i>s</i> results in energy consumption <i>s</i>^#945; per time unit. We consider the general case where each job has a monotonously increasing cost function that penalizes delay. This includes the so far considered cases of deadlines and flow time. For any type of delay cost functions, we obtain the following results: Any β-approximation algorithm for a single processor yields a randomized β<i>B</i>_α-approximation algorithm for multiple processors, where <i>B</i>_α is the αth Bell number, that is, the number of partitions of a set of size α. Analogously, we show that any β-competitive online algorithm for a single processor yields a β<i>B</i>_α-competitive online algorithm for multiple processors. Finally, we show that any β-approximation algorithm for multiple processors with migration yields a deterministic β<i>B</i>_α-approximation algorithm for multiple processors without migration. These facts improve several approximation ratios and lead to new results. For instance, we obtain the first constant factor online and offline approximation algorithm for multiple processors without migration for arbitrary release times, deadlines, and job sizes. All algorithms are based on the surprising fact that we can remove migration with a blowup of <i>B</i>_α in expectation.

Tradeoffs and Average-Case Equilibria in Selfish Routing

We study Nash equilibria in a selfish routing game on m parallel links with transmission speeds. Each player seeks to communicate a message by choosing one of the links, and each player desires to minimize his experienced transmission time (latency). For evaluating the social cost of Nash equilibria, we consider the price of anarchy, which is the largest ratio of the cost of any Nash equilibrium compared to the optimum solution. Similarly, we consider the price of stability, which is the smallest ratio. The main purpose of this article is to quantify the influence of three parameters upon the prices of the game: the total traffic in the network; restrictions of the players in terms of link choice; and fluctuations in message lengths. Our main interest is to bound the sum of all player latencies, which we refer to as collective latency. For this cost, the prices of anarchy and stability are Θ(n/t), where n is the number of players and t the sum of message lengths (total traffic); that is, Nash equilibria approximate the optimum solution up to a constant factor if the traffic is high. If each player is restricted to choose from a subset of links, these link restrictions can cause a degradation in performance of order Θ(\sqrt{m}). The prices of anarchy and stability increase to Θ(n\sqrt{m}/t). We capture fluctuations in message lengths through a stochastic model, in which we valuate Nash equilibria in terms of their expected price of anarchy. The expected price is Θ(n/\mathbb{E}[T]), where \mathbb{E}[T] is the expected traffic. The stochastic model resembles the deterministic one, even for the efficiency loss of order Θ(\sqrt{m}) for link restrictions. For the social cost function maximum latency, the (expected) price of anarchy is 1 + m2/t. In this case, Nash equilibria are almost optimal solutions for congested networks. Similar results hold when the cost function is a polynomial of the link loads.

APPROXIMATING THE JOINT REPLENISHMENT PROBLEM WITH DEADLINES

The objective of the classical Joint Replenishment Problem (JRP) is to minimize ordering costs by combining orders in two stages, first at some retailers, and then at a warehouse. These orders are needed to satisfy demands that appear over time at the retailers. We investigate the natural special case that each demand has a deadline until when it needs to be satisfied. For this case, we present a randomized 5/3-approximation algorithm. We moreover prove that JRP with deadlines is APX-hard. Finally, we extend the known hardness results by showing that JRP with linear delay cost functions is NP-hard, even if each retailer has to satisfy only three demands.

The Influence of Link Restrictions on (Random) Selfish Routing

In this paper we consider the influence of link restrictions on the price of anarchy for several social cost functions in the following model of selfish routing. Each of nplayers in a network game seeks to send a message with a certain length by choosing one of mparallel links. Each player is restricted to transmit over a certain subset of links and desires to minimize his own transmission-time (latency). We study Nash equilibria of the game, in which no player can decrease his latency by unilaterally changing his link. Our analysis of this game captures two important aspects of network traffic: the dependency of the overall network performance on the total traffic tand fluctuations in the length of the respective message-lengths. For the latter we use a probabilistic model in which message lengths are random variables. We evaluate the (expected) price of anarchy of the game for two social cost functions. For total latency cost, we show the tight result that the price of anarchy is essentially

The Bell Is Ringing in Speed-Scaled Multiprocessor Scheduling

{\it \Theta}({n\sqrt{m}/t})

Tradeoffs and average-case equilibria in selfish routing

. Hence, even for congested networks, when the traffic is linear in the number of players, Nash equilibria approximate the social optimum only by a factor of

Latency Constrained Aggregation in Chain Networks Admits a PTAS

{\it \Theta}({\sqrt{m}})

Approximation Algorithms for Generalized and Variable-Sized Bin Covering

. This efficiency loss is caused by link restrictions and remains stable even under message fluctuations, which contrasts the unrestricted case where Nash equilibria achieve a constant factor approximation. For maximum latency the price of anarchy is at most 1 + m2/t. In this case Nash equilibria can be (almost) optimal solutions for congested networks depending on the values for mand t. In addition, our analyses yield average-case analyses of a polynomial time algorithm for computing Nash equilibria in this model.

Optimal Algorithms for Train Shunting and Relaxed List Update Problems.

This paper investigates the problem of scheduling jobs on multiple speed-scaled processors, i.e., we have constant α>1 such that running a processor at speed s results in energy consumption sα per time unit. We consider the general case where each job has a monotonously increasing cost function that penalizes delay. This includes the so far considered cases of deadlines, flow time, and weighted flow time. For any type of delay cost functions, we obtain the following results: Any β-approximation algorithm for a single processor yields a randomized βBα-approximation algorithm for multiple processors, where Bα is the αth Bell number, that is, the number of partitions of a set of size α. The generated schedule is without migration, but we compare it to an optimal schedule with migration. Hence, this result holds for migratory and non-migratory schedules. Analogously, we show that any β-competitive online algorithm for a single processor yields a βBα-competitive online algorithm for multiple processors. Finally, we show that any β-approximation algorithm for multiple processors with migration yields a deterministic βBα-approximation algorithm for multiple processors without migration. These facts improve several approximation ratios and lead to new results. For instance, we obtain the first constant factor online and offline approximation algorithm for multiple processors without migration for arbitrary release times, deadlines, and job sizes.

Adversarial models in paging

We consider the price of selfish routing in terms of tradeoffs and from an average-case perspective. Each player in a network game seeks to send a message with a certain length by choosing one of several parallel links that have transmission speeds. A player desires to minimize his own transmission time (latency). We study the quality of Nash equilibria of the game, in which no player can decrease his latency by unilaterally changing his link. We treat two important aspects of network-traffic management: the influence of the total traffic upon network performance and fluctuations in the lengths of the messages. We introduce a probabilistic model where message lengths are random variables and evaluate the expected price of anarchy of the game for various social cost functions. For total latency social cost, which was only scarcely considered in previous work so far, we show that the price of anarchy is Θ(n/t), where n is the number of players and t the total message-length. The bound states that the relative quality of Nash equilibria in comparison with the social optimum increase with increasing traffic. This result also transfers to the situation when fluctuations are present, as the expected price of anarchy is O(n/)/(E[T]), where (E[T] is the expected traffic. For maximum latency the expected price of anarchy is even 1 + o(1) for sufficiently large traffic. Our results also have algorithmic implications: Our analyses of the expected prices can be seen average-case analyses of a local search algorithm that computes Nash equilibria in polynomial time.