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Split Scheduling with Uniform Setup Times
[article]

2012
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arXiv
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pre-print

We study a scheduling problem in which jobs may be split into parts, where the parts of a split job may be processed simultaneously on more than one machine. Each part of a job requires a setup time, however, on the machine where the job part is processed. During setup a machine cannot process or set up any other job. We concentrate on the basic case in which setup times are job-, machine-, and sequence-independent. Problems of this kind were encountered when modelling practical problems in

arXiv:1212.1754v1
fatcat:wuaxyweqcvfb7aqlande2w4dia
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... ning disaster relief operations. Our main algorithmic result is a polynomial-time algorithm for minimising total completion time on two parallel identical machines. We argue why the same problem with three machines is not an easy extension of the two-machine case, leaving the complexity of this case as a tantalising open problem. We give a constant-factor approximation algorithm for the general case with any number of machines and a polynomial-time approximation scheme for a fixed number of machines. For the version with objective minimising weighted total completion time we prove NP-hardness. Finally, we conclude with an overview of the state of the art for other split scheduling problems with job-, machine-, and sequence-independent setup times.##
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Scheduling over Scenarios on Two Machines
[article]

2014
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arXiv
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pre-print

AvZ was partially supported by

arXiv:1404.4766v1
fatcat:a5ufdkvpvncy3fs2mouafgjudm
*Suzann*Wilson Matthews summer research award. ...##
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A Duality Based 2-Approximation Algorithm for Maximum Agreement Forest
[article]

2018
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arXiv
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pre-print

We give a 2-approximation algorithm for the Maximum Agreement Forest problem on two rooted binary trees. This NP-hard problem has been studied extensively in the past two decades, since it can be used to compute the rooted Subtree Prune-and-Regraft (rSPR) distance between two phylogenetic trees. Our algorithm is combinatorial and its running time is quadratic in the input size. To prove the approximation guarantee, we construct a feasible dual solution for a novel linear programming

arXiv:1811.05916v1
fatcat:mwvvjovh2fgt5iz3gc35rp3vsy
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... In addition, we show this linear program is stronger than previously known formulations, and we give a compact formulation, showing that it can be solved in polynomial time##
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Approximating Real-Time Scheduling on Identical Machines
[chapter]

2014
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Lecture Notes in Computer Science
*

We study the problem of assigning n tasks to m identical parallel machines in the real-time scheduling setting, where each task recurrently releases jobs that must be completed by their deadlines. The goal is to find a partition of the task set over the machines such that each job that is released by a task can meet its deadline. Since this problem is co-NP-hard, the focus is on finding α-approximation algorithms in the resource augmentation setting, i.e., finding a feasible partition on

doi:10.1007/978-3-642-54423-1_48
fatcat:zlkmukgscvdddp2nuwoe5qbmmq
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... s running at speed α ≥ 1, if some feasible partition exists on unit-speed machines. Recently, Chen and Chakraborty gave a polynomial-time approximation scheme if the ratio of the largest to the smallest relative deadline of the tasks, λ, is bounded by a constant. However, their algorithm has a super-exponential dependence on λ and hence does not extend to larger values of λ. Our main contribution is to design an approximation scheme with a substantially improved running-time dependence on λ. In particular, our algorithm depends exponentially on log λ and hence has quasi-polynomial running time even if λ is polynomially bounded. This improvement is based on exploiting various structural properties of approximate demand bound functions in different ways, which might be of independent interest. Supported by the NWO VIDI grant 639.022.211. system, such that all jobs from all tasks meet their deadlines. In this paper, we consider the feasibility question of scheduling a set of sporadic tasks to multiple identical machines (processors). This problem and related problems in real-time scheduling have received great attention in the last years; see for example [1, 4, 6] and the references therein. Single-processor case: Determining the feasibility of a task system on a single (preemptive 4 ) processor is quite well-understood. It is well-known that the hardest case for feasibility is when the first jobs of all tasks arrive simultaneously and all subsequent jobs arrive as rapidly as legally possible [5] . That is, we can assume that for each task τ in the task system, the jobs of τ arrive at times 0, p τ , 2p τ , . . .. This sequence of job-arrivals is called the synchronous arrival sequence. Another well-known fact [11] is that the Earliest Deadline First (EDF) algorithm, that schedules at any time the job with the earliest absolute deadline, will always produce a valid schedule for any sequence of jobs that is feasible. Although one can validate whether a task system is feasible by running EDF, this does not provide an efficient polynomial-time feasibility test. The problem is that the periodic nature of jobs together with their relative deadlines can introduce complicated long-range dependencies. In particular, the infeasibility may occur only at a very late time in the schedule, say close to the hyperperiod (which is the least common multiple of the periods of the tasks). In fact, no polynomial-time feasibility test on a uniprocessor is likely to exist, unless P=co-NP [10] . For more results on scheduling sporadic task systems on a single processor, we refer to Baruah and Goosens [4]. Multiprocessor case: For multiprocessor systems, there are two main paradigms for scheduling: global vs. partitioned scheduling. In partitioned scheduling each task is assigned to one of the machines and all jobs corresponding to this task must be scheduled on that machine. In global scheduling, tasks can use all machines and jobs can even be migrated. Partitioned scheduling is used much more than global scheduling as it is easier to implement and has no communication overhead, which is required if a single task is split between multiple processors. The communication may also lead to security issues. In this paper, we only consider partitioned scheduling. Observe that in this setting, given a partition of the tasks over the machines, determining its feasibility simply reduces to several independent uniprocessor feasibility problems -one for each machine. Together with the facts for uniprocessor feasibility, the problem we study can be viewed as follows: Find a partition of tasks among machines, such that for each machine, the synchronous arrival sequence for tasks assigned to that machine is feasible for EDF. Clearly, this problem is also co-NP-hard and, as we shall see, it is also NP-hard. Resource Augmentation and α-feasibility: The hardness of the problem leads us to finding a good approximation algorithm. As usual, we consider the resource##
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The traveling salesman problem on cubic and subcubic graphs
[article]

2011
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arXiv
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pre-print

We study the Travelling Salesman Problem (TSP) on the metric completion of cubic and subcubic graphs, which is known to be NP-hard. The problem is of interest because of its relation to the famous 4/3 conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal values of the TSP and its linear programming relaxation (the subtour elimination relaxation), is 4/3. We present the first algorithm for cubic graphs with approximation ratio 4/3. The

arXiv:1107.1052v1
fatcat:sbklbt5rebaytnvbpsd7atmd7q
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... f uses polyhedral techniques in a surprising way, which is of independent interest. In fact we prove constructively that for any cubic graph on n vertices a tour of length 4n/3-2 exists, which also implies the 4/3 conjecture, as an upper bound, for this class of graph-TSP. Recently, Mömke and Svensson presented a randomized algorithm that gives a 1.461-approximation for graph-TSP on general graphs and as a side result a 4/3-approximation algorithm for this problem on subcubic graphs, also settling the 4/3 conjecture for this class of graph-TSP. We will present a way to derandomize their algorithm which leads to a smaller running time than the obvious derandomization. All of the latter also works for multi-graphs.##
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Split scheduling with uniform setup times

2014
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Journal of Scheduling
*

We encountered such problems in studying disaster relief operations (

doi:10.1007/s10951-014-0370-4
fatcat:l6i7oow5pjcg5i4lklypu7mw2u
*Van**der**Ster*2010). ... P | pmtn | w j C j NP-hard (Bruno et al. 1974 ) P T A S ( Afrati et al. 1999) P | s, split | C max NP-hard (*Van**der**Ster*2010) [cf. ...##
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Assigning sporadic tasks to unrelated machines

2014
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Mathematical programming
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We study the problem of assigning sporadic tasks to unrelated machines such that the tasks on each machine can be feasibly scheduled. Despite its importance for modern real-time systems, this problem has not been studied before. We present a polynomial-time algorithm which approximates the problem with a constant speedup factor of 8 + 2 √ 6 ≈ 12.9 and show that any polynomial-time algorithm needs a speedup factor of at least 2, unless P = NP. In the case of a constant number of machines we give

doi:10.1007/s10107-014-0786-9
fatcat:dyxvmgapzbbiznaywqe2qfebma
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... a polynomial-time approximation scheme. Key to these results are two new relaxations of the demand bound function, the function that yields a sufficient and necessary condition for a task system on a single machine to be feasible. In particular, we present new methods to approximate this function to obtain useful structural properties while incurring only bounded loss in the approximation quality. For the constant speedup result we employ a very general rounding procedure for linear programs (LPs) which model assignment problems with capacity-type constraints. It ensures that the cost of the rounded integral solution is no more than the cost of the optimal fractional LP solution and the capacity constraints are violated only by a bounded factor, depending on the structure of the matrix that defines the LP. In fact, our rounding scheme generalizes the well-known 2-approximation algorithm for the generalized assignment problem due to Shmoys and Tardos.##
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TSP on Cubic and Subcubic Graphs
[chapter]

2011
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Lecture Notes in Computer Science
*

We study the Travelling Salesman Problem (TSP) on the metric completion of cubic and subcubic graphs, which is known to be NP-hard. The problem is of interest because of its relation to the famous 4/3 conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal values of the TSP and its linear programming relaxation, is 4/3. Using polyhedral techniques in an interesting way, we obtain a polynomial-time 4/3-approximation algorithm for this

doi:10.1007/978-3-642-20807-2_6
fatcat:donyaxftkfcmpd44bbthuyueay
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... on cubic graphs, improving upon Christofides' 3/2-approximation, and upon the 3/2 − 5/389 ≈ 1.487-approximation ratio by Gamarnik, Lewenstein and Svirdenko for the case the graphs are also 3-edge connected. We also prove that, as an upper bound, the 4/3 conjecture is true for this problem on cubic graphs. For subcubic graphs we obtain a polynomial-time 7/5-approximation algorithm and a 7/5 bound on the integrality gap.##
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A Duality Based 2-Approximation Algorithm for Maximum Agreement Forest
[article]

2016
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arXiv
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pre-print

AvZ was supported in part by a William & Mary Summer Research Award, NSF Prime Award: HRD-1107147, Women in Scientific Education (WISE) and by a grant from the Simons Foundation (#359525, Anke

arXiv:1511.06000v2
fatcat:bd6aaxqpgfcalie76pq32vchqe
*Van*Zuylen ...##
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Scheduling over Scenarios on Two Machines
[chapter]

2014
*
Lecture Notes in Computer Science
*

AvZ was partially supported by

doi:10.1007/978-3-319-08783-2_48
fatcat:ft5vevzunjerdlby742zponhdq
*Suzann*Wilson Matthews summer research award. ...##
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The traveling salesman problem on cubic and subcubic graphs

2012
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Mathematical programming
*

We study the traveling salesman problem (TSP) on the metric completion of cubic and subcubic graphs, which is known to be NP-hard. The problem is of interest because of its relation to the famous 4/3-conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal value of a TSP instance and that of its linear programming relaxation (the subtour elimination relaxation), is 4/3. We present the first algorithm for cubic graphs with approximation ratio

doi:10.1007/s10107-012-0620-1
fatcat:34lq2cwdbrfqzntudb7v5t2q3u
## more »

... 4/3. The proof uses polyhedral techniques in a surprising way, which is of independent interest. In fact we prove constructively that for any cubic graph on n vertices a tour of length 4n/3 − 2 exists, which also implies the 4/3-conjecture, as an upper bound, for this class of graph-TSP. Recently, Mömke and Svensson presented an algorithm##
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Minimizing worst-case and average-case makespan over scenarios

2016
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Journal of Scheduling
*

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 scenarios in an explicitly given set. Each scenario is a subset of jobs that must be executed in that scenario. 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

doi:10.1007/s10951-016-0484-y
fatcat:lixywuhpvjfg3cdbwzh2iertje
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... roximability. We also consider some (easier) special cases. Combinatorial optimization problems under scenarios in general, and scheduling problems under scenarios in particular, have seen only limited research attention so far. With this paper, we make a step in this interesting research direction.##
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Mixed-Criticality Scheduling of Sporadic Task Systems
[chapter]

2011
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Lecture Notes in Computer Science
*

We consider the scheduling of mixed-criticality task systems, that is, systems where each task to be scheduled has multiple levels of worst-case execution time estimates. We design a scheduling algorithm, EDF-VD, whose effectiveness we analyze using the processor speedup metric: we show that any 2-level task system that is schedulable on a unit-speed processor is correctly scheduled by EDF-VD using speed φ; here φ < 1.619 is the golden ratio. We also show how to generalize the algorithm to K >

doi:10.1007/978-3-642-23719-5_47
fatcat:zb42weoqbrcu3k5zsonn4qkc74
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... criticality levels.We finally consider 2-level instances on m identical machines. We prove speedup bounds for scheduling an independent collection of jobs and for the partitioned scheduling of a 2-level task system. 2##
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Preemptive Uniprocessor Scheduling of Mixed-Criticality Sporadic Task Systems

2015
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Journal of the ACM
*

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A Duality Based 2-Approximation Algorithm for Maximum Agreement Forest * †

unpublished

We give a 2-approximation algorithm for the Maximum Agreement Forest problem on two rooted binary trees. This NP-hard problem has been studied extensively in the past two decades, since it can be used to compute the Subtree Prune-and-Regraft (SPR) distance between two phylogenetic trees. Our result improves on the very recent 2.5-approximation algorithm due to Shi, Feng, You and Wang (2015). Our algorithm is the first approximation algorithm for this problem that uses LP duality in its

fatcat:zrq2bikqq5dd7eyrtp4x7dt43a
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... 1998 ACM Subject Classification F.2.2 Analysis of algorithms and problem complexity
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