36,603 research outputs found
Parametrisierte Algorithmen für Ganzzahlige Lineare Programme und deren Anwendungen für Zuweisungsprobleme
This thesis is concerned with solving NP-hard problems. We consider two prominent strategies of coping with such computationally hard questions efficiently. The first approach aims to design approximation algorithms, that is, we are content to find good, but non-optimal solutions in polynomial time. The second strategy is called Fixed-Parameter Tractability (FPT) and considers parameters of the instance to capture the hardness of the problem and by that, obtain efficient algorithms with respect to the remaining input. This thesis employs both strategies jointly to develop efficient approximation and exact algorithms using parameterization and modeling the problem as structured integer linear programs (ILPs), which can be solved in FPT. In the first part of this work, we concentrate on these well-structured ILPs. On the one hand, we develop an efficient algorithm for block-structured integer linear programs called n-fold ILPs. On the other hand, we investigate the similarly block-structured 2-stage stochastic ILPs and prove conditional lower bounds regarding the running time of any algorithm solving them that match the best known upper bounds. We also prove the tightness of certain structural parameters called sensitivity and proximity for ILPs which arise from combinatorial questions such as allocation problems. The second part utilizes n-fold ILPs and structural properties to add to and improve upon known results for Scheduling and Bin Packing problems. We design exact FPT algorithms for the Scheduling With Clique Incompatibilities, Bin Packing, and Multiple Knapsack problems. Further, we provide constant-factor approximation algorithms and polynomial time approximation schemes (PTAS) for the Class Constraint Scheduling problems. Broadening our scope, we also investigate this problem and the closely related Cardinality Constraint Scheduling problem in the online setting and derive lower bounds for the approximation ratios as well as a PTAS for them. Altogether, this thesis contributes to the knowledge about structured ILPs, proves their limits and reaffirms their usefulness for a plethora of allocation problems. In doing so, various new and improved algorithms with respect to the running time or approximation quality emerge
Stochastic and Robust Scheduling in the Cloud
Users of cloud computing services are offered rapid access to computing resources via the Internet.
Cloud providers use different pricing options such as (i) time slot reservation in advance at a fixed
price and (ii) on-demand service at a (hourly) pay-as-used basis. Choosing the best combination
of pricing options is a challenging task for users, in particular, when the instantiation of computing
jobs underlies uncertainty.
We propose a natural model for two-stage scheduling under uncertainty that captures such
resource provisioning and scheduling problem in the cloud. Reserving a time unit for processing
jobs incurs some cost, which depends on when the reservation is made: a priori decisions, based
only on distributional information, are much cheaper than on-demand decisions when the actual
scenario is known. We consider both stochastic and robust versions of scheduling unrelated
machines with objectives of minimizing the sum of weighted completion times Pj wjCj and the makespan maxj Cj . Our main contribution is an (8+)-approximation algorithm for the min-sum
objective for the stochastic polynomial-scenario model. The same technique gives a (7.11 + )-
approximation for minimizing the makespan. The key ingredient is an LP-based separation
of jobs and time slots to be considered in either the first or the second stage only, and then
approximately solving the separated problems. At the expense of another our results hold for
any arbitrary scenario distribution given by means of a black-box. Our techniques also yield
approximation algorithms for robust two-stage scheduling
Some complexity and approximation results for coupled-tasks scheduling problem according to topology
We consider the makespan minimization coupled-tasks problem in presence of
compatibility constraints with a specified topology. In particular, we focus on
stretched coupled-tasks, i.e. coupled-tasks having the same sub-tasks execution
time and idle time duration. We study several problems in framework of classic
complexity and approximation for which the compatibility graph is bipartite
(star, chain,. . .). In such a context, we design some efficient
polynomial-time approximation algorithms for an intractable scheduling problem
according to some parameters
Two Timescale Convergent Q-learning for Sleep--Scheduling in Wireless Sensor Networks
In this paper, we consider an intrusion detection application for Wireless
Sensor Networks (WSNs). We study the problem of scheduling the sleep times of
the individual sensors to maximize the network lifetime while keeping the
tracking error to a minimum. We formulate this problem as a
partially-observable Markov decision process (POMDP) with continuous
state-action spaces, in a manner similar to (Fuemmeler and Veeravalli [2008]).
However, unlike their formulation, we consider infinite horizon discounted and
average cost objectives as performance criteria. For each criterion, we propose
a convergent on-policy Q-learning algorithm that operates on two timescales,
while employing function approximation to handle the curse of dimensionality
associated with the underlying POMDP. Our proposed algorithm incorporates a
policy gradient update using a one-simulation simultaneous perturbation
stochastic approximation (SPSA) estimate on the faster timescale, while the
Q-value parameter (arising from a linear function approximation for the
Q-values) is updated in an on-policy temporal difference (TD) algorithm-like
fashion on the slower timescale. The feature selection scheme employed in each
of our algorithms manages the energy and tracking components in a manner that
assists the search for the optimal sleep-scheduling policy. For the sake of
comparison, in both discounted and average settings, we also develop a function
approximation analogue of the Q-learning algorithm. This algorithm, unlike the
two-timescale variant, does not possess theoretical convergence guarantees.
Finally, we also adapt our algorithms to include a stochastic iterative
estimation scheme for the intruder's mobility model. Our simulation results on
a 2-dimensional network setting suggest that our algorithms result in better
tracking accuracy at the cost of only a few additional sensors, in comparison
to a recent prior work
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