331 research outputs found
Polynomial-time approximation schemes for scheduling problems with time lags
We identify two classes of machine scheduling problems with time lags that possess Polynomial-Time Approximation Schemes (PTASs). These classes together, one for minimizing makespan and one for minimizing total completion time, include many well-studied time lag scheduling problems. The running times of these approximation schemes are polynomial in the number of jobs, but exponential in the number of machines and the ratio between the largest time lag and the smallest positive operation time. These classes constitute the first PTAS results for scheduling problems with time lags
Lift-and-Round to Improve Weighted Completion Time on Unrelated Machines
We consider the problem of scheduling jobs on unrelated machines so as to
minimize the sum of weighted completion times. Our main result is a
-approximation algorithm for some fixed , improving upon the
long-standing bound of 3/2 (independently due to Skutella, Journal of the ACM,
2001, and Sethuraman & Squillante, SODA, 1999). To do this, we first introduce
a new lift-and-project based SDP relaxation for the problem. This is necessary
as the previous convex programming relaxations have an integrality gap of
. Second, we give a new general bipartite-rounding procedure that produces
an assignment with certain strong negative correlation properties.Comment: 21 pages, 4 figure
Single machine scheduling with job-dependent machine deterioration
We consider the single machine scheduling problem with job-dependent machine
deterioration. In the problem, we are given a single machine with an initial
non-negative maintenance level, and a set of jobs each with a non-preemptive
processing time and a machine deterioration. Such a machine deterioration
quantifies the decrement in the machine maintenance level after processing the
job. To avoid machine breakdown, one should guarantee a non-negative
maintenance level at any time point; and whenever necessary, a maintenance
activity must be allocated for restoring the machine maintenance level. The
goal of the problem is to schedule the jobs and the maintenance activities such
that the total completion time of jobs is minimized. There are two variants of
maintenance activities: in the partial maintenance case each activity can be
allocated to increase the machine maintenance level to any level not exceeding
the maximum; in the full maintenance case every activity must be allocated to
increase the machine maintenance level to the maximum. In a recent work, the
problem in the full maintenance case has been proven NP-hard; several special
cases of the problem in the partial maintenance case were shown solvable in
polynomial time, but the complexity of the general problem is left open. In
this paper we first prove that the problem in the partial maintenance case is
NP-hard, thus settling the open problem; we then design a -approximation
algorithm.Comment: 15 page
Optimal Algorithms for Scheduling under Time-of-Use Tariffs
We consider a natural generalization of classical scheduling problems in
which using a time unit for processing a job causes some time-dependent cost
which must be paid in addition to the standard scheduling cost. We study the
scheduling objectives of minimizing the makespan and the sum of (weighted)
completion times. It is not difficult to derive a polynomial-time algorithm for
preemptive scheduling to minimize the makespan on unrelated machines. The
problem of minimizing the total (weighted) completion time is considerably
harder, even on a single machine. We present a polynomial-time algorithm that
computes for any given sequence of jobs an optimal schedule, i.e., the optimal
set of time-slots to be used for scheduling jobs according to the given
sequence. This result is based on dynamic programming using a subtle analysis
of the structure of optimal solutions and a potential function argument. With
this algorithm, we solve the unweighted problem optimally in polynomial time.
For the more general problem, in which jobs may have individual weights, we
develop a polynomial-time approximation scheme (PTAS) based on a dual
scheduling approach introduced for scheduling on a machine of varying speed. As
the weighted problem is strongly NP-hard, our PTAS is the best possible
approximation we can hope for.Comment: 17 pages; A preliminary version of this paper with a subset of
results appeared in the Proceedings of MFCS 201
Scheduling over Scenarios on Two Machines
We consider scheduling problems over scenarios where the goal is to find a
single assignment of the jobs to the machines which performs well over all
possible scenarios. Each scenario is a subset of jobs that must be executed in
that scenario and all scenarios are given explicitly. The two objectives that
we consider are minimizing the maximum makespan over all scenarios and
minimizing the sum of the makespans of all scenarios. For both versions, we
give several approximation algorithms and lower bounds on their
approximability. With this research into optimization problems over scenarios,
we have opened a new and rich field of interesting problems.Comment: To appear in COCOON 2014. The final publication is available at
link.springer.co
A PTAS for Minimizing Average Weighted Completion Time With Release Dates on Uniformly Related Machines
A classical scheduling problem is to find schedules that minimize average weighted completion time of jobs with release dates. When multiple machines are available, the machine environments may range from identical machines (the processing time required by a job is invariant across the machines) at one end, to unrelated machines (the processing time required by a job on any machine is an arbitrary function of the specific machine) at the other end of the spectrum. While the problem is strongly NP-hard even in the case of a single machine, constant factor approximation algorithms have been known for even the most general machine environment of unrelated machines. Recently, a polynomial-time approximation scheme (PTAS) was discovered for the case of identical parallel machines [1]. In contrast, it is known that this problem is MAX SNP-hard for unrelated machines [10]. An important open problem is to determine the approximability of the intermediate case of uniformly related machines where each machine i has a speed si and it takes p/si time to executing a job of processing size pIn this paper, we resolve this problem by obtaining a PTAS for the problem. This improves the earlier known ratio of (2 + ∈) for the problem
From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz
The next few years will be exciting as prototype universal quantum processors
emerge, enabling implementation of a wider variety of algorithms. Of particular
interest are quantum heuristics, which require experimentation on quantum
hardware for their evaluation, and which have the potential to significantly
expand the breadth of quantum computing applications. A leading candidate is
Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates
between applying a cost-function-based Hamiltonian and a mixing Hamiltonian.
Here, we extend this framework to allow alternation between more general
families of operators. The essence of this extension, the Quantum Alternating
Operator Ansatz, is the consideration of general parametrized families of
unitaries rather than only those corresponding to the time-evolution under a
fixed local Hamiltonian for a time specified by the parameter. This ansatz
supports the representation of a larger, and potentially more useful, set of
states than the original formulation, with potential long-term impact on a
broad array of application areas. For cases that call for mixing only within a
desired subspace, refocusing on unitaries rather than Hamiltonians enables more
efficiently implementable mixers than was possible in the original framework.
Such mixers are particularly useful for optimization problems with hard
constraints that must always be satisfied, defining a feasible subspace, and
soft constraints whose violation we wish to minimize. More efficient
implementation enables earlier experimental exploration of an alternating
operator approach to a wide variety of approximate optimization, exact
optimization, and sampling problems. Here, we introduce the Quantum Alternating
Operator Ansatz, lay out design criteria for mixing operators, detail mappings
for eight problems, and provide brief descriptions of mappings for diverse
problems.Comment: 51 pages, 2 figures. Revised to match journal pape
The robust single machine scheduling problem with uncertain release and processing times
In this work, we study the single machine scheduling problem with uncertain
release times and processing times of jobs. We adopt a robust scheduling
approach, in which the measure of robustness to be minimized for a given
sequence of jobs is the worst-case objective function value from the set of all
possible realizations of release and processing times. The objective function
value is the total flow time of all jobs. We discuss some important properties
of robust schedules for zero and non-zero release times, and illustrate the
added complexity in robust scheduling given non-zero release times. We propose
heuristics based on variable neighborhood search and iterated local search to
solve the problem and generate robust schedules. The algorithms are tested and
their solution performance is compared with optimal solutions or lower bounds
through numerical experiments based on synthetic data
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