201 research outputs found
A PTAS for Minimizing Weighted Completion Time on Uniformly Related Machines (Extended Abstract)
We consider the well known problem of scheduling jobs with release dates to minimize their average weighted completion time. When multiple machines are available, the machine environment may range from identical machines (the processing time required by a job is invariant across the machines) at one end of the spectrum to unrelated machines (the processing time required by a job on each machine is specified by an arbitrary vector) at the other end. While the problem is strongly NP-hard even in the case of a single machine, constant factor approximation algorithms are known for even the most general machine environment of unrelated machines. Recently a PTAS was discovered for the case of identical parallel machines [1]. In contrast, the problem is MAX SNP-hard for unrelated machines [11]. An important open problem was to determine the approximability of the intermediate case of uniformly related machines where each machine has a speed and it takes p/s time to process a job of size p on a machine with speed s. We resolve the complexity of this problem by obtaining a PTAS. This improves the earlier known approximation ratio of (2 + ε)
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
Malleable Scheduling Beyond Identical Machines
In malleable job scheduling, jobs can be executed simultaneously on multiple machines with the processing time depending on the number of allocated machines. Jobs are required to be executed non-preemptively and in unison, in the sense that they occupy, during their execution, the same time interval over all the machines of the allocated set. In this work, we study generalizations of malleable job scheduling inspired by standard scheduling on unrelated machines. Specifically, we introduce a general model of malleable job scheduling, where each machine has a (possibly different) speed for each job, and the processing time of a job j on a set of allocated machines S depends on the total speed of S for j. For machines with unrelated speeds, we show that the optimal makespan cannot be approximated within a factor less than e/(e-1), unless P = NP. On the positive side, we present polynomial-time algorithms with approximation ratios 2e/(e-1) for machines with unrelated speeds, 3 for machines with uniform speeds, and 7/3 for restricted assignments on identical machines. Our algorithms are based on deterministic LP rounding and result in sparse schedules, in the sense that each machine shares at most one job with other machines. We also prove lower bounds on the integrality gap of 1+phi for unrelated speeds (phi is the golden ratio) and 2 for uniform speeds and restricted assignments. To indicate the generality of our approach, we show that it also yields constant factor approximation algorithms (i) for minimizing the sum of weighted completion times; and (ii) a variant where we determine the effective speed of a set of allocated machines based on the L_p norm of their speeds
Better Unrelated Machine Scheduling for Weighted Completion Time via Random Offsets from Non-Uniform Distributions
In this paper we consider the classic scheduling problem of minimizing total
weighted completion time on unrelated machines when jobs have release times,
i.e, using the three-field notation. For this
problem, a 2-approximation is known based on a novel convex programming (J. ACM
2001 by Skutella). It has been a long standing open problem if one can improve
upon this 2-approximation (Open Problem 8 in J. of Sched. 1999 by Schuurman and
Woeginger). We answer this question in the affirmative by giving a
1.8786-approximation. We achieve this via a surprisingly simple linear
programming, but a novel rounding algorithm and analysis. A key ingredient of
our algorithm is the use of random offsets sampled from non-uniform
distributions.
We also consider the preemptive version of the problem, i.e, . We again use the idea of sampling offsets from non-uniform
distributions to give the first better than 2-approximation for this problem.
This improvement also requires use of a configuration LP with variables for
each job's complete schedules along with more careful analysis. For both
non-preemptive and preemptive versions, we break the approximation barrier of 2
for the first time.Comment: 24 pages. To apper in FOCS 201
Scheduling to Minimize Total Weighted Completion Time via Time-Indexed Linear Programming Relaxations
We study approximation algorithms for scheduling problems with the objective
of minimizing total weighted completion time, under identical and related
machine models with job precedence constraints. We give algorithms that improve
upon many previous 15 to 20-year-old state-of-art results. A major theme in
these results is the use of time-indexed linear programming relaxations. These
are natural relaxations for their respective problems, but surprisingly are not
studied in the literature.
We also consider the scheduling problem of minimizing total weighted
completion time on unrelated machines. The recent breakthrough result of
[Bansal-Srinivasan-Svensson, STOC 2016] gave a -approximation for the
problem, based on some lift-and-project SDP relaxation. Our main result is that
a -approximation can also be achieved using a natural and
considerably simpler time-indexed LP relaxation for the problem. We hope this
relaxation can provide new insights into the problem
Scheduling Bidirectional Traffic on a Path
We study the fundamental problem of scheduling bidirectional traffic along a
path composed of multiple segments. The main feature of the problem is that
jobs traveling in the same direction can be scheduled in quick succession on a
segment, while jobs in opposing directions cannot cross a segment at the same
time. We show that this tradeoff makes the problem significantly harder than
the related flow shop problem, by proving that it is NP-hard even for identical
jobs. We complement this result with a PTAS for a single segment and
non-identical jobs. If we allow some pairs of jobs traveling in different
directions to cross a segment concurrently, the problem becomes APX-hard even
on a single segment and with identical jobs. We give polynomial algorithms for
the setting with restricted compatibilities between jobs on a single and any
constant number of segments, respectively
How Unsplittable-Flow-Covering helps Scheduling with Job-Dependent Cost Functions
Generalizing many well-known and natural scheduling problems, scheduling with
job-specific cost functions has gained a lot of attention recently. In this
setting, each job incurs a cost depending on its completion time, given by a
private cost function, and one seeks to schedule the jobs to minimize the total
sum of these costs. The framework captures many important scheduling objectives
such as weighted flow time or weighted tardiness. Still, the general case as
well as the mentioned special cases are far from being very well understood
yet, even for only one machine. Aiming for better general understanding of this
problem, in this paper we focus on the case of uniform job release dates on one
machine for which the state of the art is a 4-approximation algorithm. This is
true even for a special case that is equivalent to the covering version of the
well-studied and prominent unsplittable flow on a path problem, which is
interesting in its own right. For that covering problem, we present a
quasi-polynomial time -approximation algorithm that yields an
-approximation for the above scheduling problem. Moreover, for
the latter we devise the best possible resource augmentation result regarding
speed: a polynomial time algorithm which computes a solution with \emph{optimal
}cost at speedup. Finally, we present an elegant QPTAS for the
special case where the cost functions of the jobs fall into at most
many classes. This algorithm allows the jobs even to have up to many
distinct release dates.Comment: 2 pages, 1 figur
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