26,817 research outputs found
Optimal on-line flow time with resource augmentation
AbstractWe study the problem of scheduling n jobs that arrive over time. We consider a non-preemptive setting on a single machine. The goal is to minimize the total flow time. We use extra resource competitive analysis: an optimal off-line algorithm which schedules jobs on a single machine is compared to a more powerful on-line algorithm that has ℓ machines. We design an algorithm of competitive ratio 1+2min(Δ1/ℓ,n1/ℓ), where Δ is the maximum ratio between two job sizes, and provide a lower bound which shows that the algorithm is optimal up to a constant factor for any constant ℓ. The algorithm works for a hard version of the problem where the sizes of the smallest and the largest jobs are not known in advance, only Δ and n are known. This gives a trade-off between the resource augmentation and the competitive ratio.We also consider scheduling on parallel identical machines. In this case the optimal off-line algorithm has m machines and the on-line algorithm has ℓm machines. We give a lower bound for this case. Next, we give lower bounds for algorithms using resource augmentation on the speed. Finally, we consider scheduling with hard deadlines, and scheduling so as to minimize the total completion time
New results on flow time with resource augmentation
We study the problem of scheduling jobs that arrive over time. We consider a non-preemptive setting on a single machine. The goal is to minimize the total flow time. We use extra resource competitive analysis: an optimal off-line algorithm which schedules jobs on a single machine is compared to a more powerful on-line algorithm that has machines. We design an algorithm of competitive ratio O(min(Delta^{1/l,n^{1/l)), where is the maximum ratio between two job sizes, and provide a lower bound which shows that the algorithm is optimal up to a constant factor for any constant . The algorithm works for a hard version of the problem where the sizes of the smallest and the largest jobs are not known in advance, only is known. This gives a trade-off between the resource augmentation and the competitive ratio. We also consider scheduling on parallel identical machines. In this case the optimal off-line algorithm has machines and the on-line algorithm has machines. We give a lower bound for this case. Next, we give lower bounds for algorithms using resource augmentation on the speed. Finally, we consider scheduling with hard deadlines
Minimizing Flow Time in the Wireless Gathering Problem
We address the problem of efficient data gathering in a wireless network
through multi-hop communication. We focus on the objective of minimizing the
maximum flow time of a data packet. We prove that no polynomial time algorithm
for this problem can have approximation ratio less than \Omega(m^{1/3) when
packets have to be transmitted, unless . We then use resource
augmentation to assess the performance of a FIFO-like strategy. We prove that
this strategy is 5-speed optimal, i.e., its cost remains within the optimal
cost if we allow the algorithm to transmit data at a speed 5 times higher than
that of the optimal solution we compare to
Approximating k-Forest with Resource Augmentation: A Primal-Dual Approach
In this paper, we study the -forest problem in the model of resource
augmentation. In the -forest problem, given an edge-weighted graph ,
a parameter , and a set of demand pairs , the
objective is to construct a minimum-cost subgraph that connects at least
demands. The problem is hard to approximate---the best-known approximation
ratio is . Furthermore, -forest is as hard to
approximate as the notoriously-hard densest -subgraph problem.
While the -forest problem is hard to approximate in the worst-case, we
show that with the use of resource augmentation, we can efficiently approximate
it up to a constant factor.
First, we restate the problem in terms of the number of demands that are {\em
not} connected. In particular, the objective of the -forest problem can be
viewed as to remove at most demands and find a minimum-cost subgraph that
connects the remaining demands. We use this perspective of the problem to
explain the performance of our algorithm (in terms of the augmentation) in a
more intuitive way.
Specifically, we present a polynomial-time algorithm for the -forest
problem that, for every , removes at most demands and has
cost no more than times the cost of an optimal algorithm
that removes at most demands
A Constant Factor Approximation Algorithm for Unsplittable Flow on Paths
In the unsplittable flow problem on a path, we are given a capacitated path
and tasks, each task having a demand, a profit, and start and end
vertices. The goal is to compute a maximum profit set of tasks, such that for
each edge of , the total demand of selected tasks that use does not
exceed the capacity of . This is a well-studied problem that has been
studied under alternative names, such as resource allocation, bandwidth
allocation, resource constrained scheduling, temporal knapsack and interval
packing.
We present a polynomial time constant-factor approximation algorithm for this
problem. This improves on the previous best known approximation ratio of
. The approximation ratio of our algorithm is for any
.
We introduce several novel algorithmic techniques, which might be of
independent interest: a framework which reduces the problem to instances with a
bounded range of capacities, and a new geometrically inspired dynamic program
which solves a special case of the maximum weight independent set of rectangles
problem to optimality. In the setting of resource augmentation, wherein the
capacities can be slightly violated, we give a -approximation
algorithm. In addition, we show that the problem is strongly NP-hard even if
all edge capacities are equal and all demands are either~1,~2, or~3.Comment: 37 pages, 5 figures Version 2 contains the same results as version 1,
but the presentation has been greatly revised and improved. References have
been adde
SELFISHMIGRATE: A Scalable Algorithm for Non-clairvoyantly Scheduling Heterogeneous Processors
We consider the classical problem of minimizing the total weighted flow-time
for unrelated machines in the online \emph{non-clairvoyant} setting. In this
problem, a set of jobs arrive over time to be scheduled on a set of
machines. Each job has processing length , weight , and is
processed at a rate of when scheduled on machine . The online
scheduler knows the values of and upon arrival of the job,
but is not aware of the quantity . We present the {\em first} online
algorithm that is {\em scalable} ((1+\eps)-speed
-competitive for any constant \eps > 0) for the
total weighted flow-time objective. No non-trivial results were known for this
setting, except for the most basic case of identical machines. Our result
resolves a major open problem in online scheduling theory. Moreover, we also
show that no job needs more than a logarithmic number of migrations. We further
extend our result and give a scalable algorithm for the objective of minimizing
total weighted flow-time plus energy cost for the case of unrelated machines
and obtain a scalable algorithm. The key algorithmic idea is to let jobs
migrate selfishly until they converge to an equilibrium. Towards this end, we
define a game where each job's utility which is closely tied to the
instantaneous increase in the objective the job is responsible for, and each
machine declares a policy that assigns priorities to jobs based on when they
migrate to it, and the execution speeds. This has a spirit similar to
coordination mechanisms that attempt to achieve near optimum welfare in the
presence of selfish agents (jobs). To the best our knowledge, this is the first
work that demonstrates the usefulness of ideas from coordination mechanisms and
Nash equilibria for designing and analyzing online algorithms
Beyond Biomass: Valuing Genetic Diversity in Natural Resource Management
Strategies for increasing production of goods from working and natural systems have raised concerns that the diversity of species on which these services depend may be eroding. This loss of natural capital threatens to homogenize global food supplies and compromise the stability of human welfare. We assess the trade off between artificial augmentation of biomass and degradation of biodiversity underlying a populations' ability to adapt to shocks. Our application involves the augmentation of wild stocks of salmon. Practices in this system have generated warnings that genetic erosion may lead to a loss of the “portfolio effect” and the value of this loss is not accounted for in decision making. We construct an integrated bioeconomic model of salmon biomass and genetic diversity. Our results show how practices that homogenize natural systems can still generate positive returns. However, the substitution of more physical capital and labor for natural capital must be maintained for gains to persist, weakens the capacity for adaptation should this investment cease, and can cause substantial loss of population wildness. We apply an emerging optimization method—approximate dynamic programming—to solve the model without simplifying restrictions imposed previously
- …