225 research outputs found
No Polynomial Kernels for Knapsack
This paper focuses on kernelization algorithms for the fundamental Knapsack
problem. A kernelization algorithm (or kernel) is a polynomial-time reduction
from a problem onto itself, where the output size is bounded by a function of
some problem-specific parameter. Such algorithms provide a theoretical model
for data reduction and preprocessing and are central in the area of
parameterized complexity. In this way, a kernel for Knapsack for some parameter
reduces any instance of Knapsack to an equivalent instance of size at most
in polynomial time, for some computable function . When
then we call such a reduction a polynomial kernel.
Our study focuses on two natural parameters for Knapsack: The number of
different item weights , and the number of different item profits
. Our main technical contribution is a proof showing that Knapsack does
not admit a polynomial kernel for any of these two parameters under standard
complexity-theoretic assumptions. Our proof discovers an elaborate application
of the standard kernelization lower bound framework, and develops along the way
novel ideas that should be useful for other problems as well. We complement our
lower bounds by showing the Knapsack admits a polynomial kernel for the
combined parameter
Scheduling Lower Bounds via AND Subset Sum
Given instances of Subset Sum, the AND
Subset Sum problem asks to determine whether all of these instances are
yes-instances; that is, whether each set of integers has a subset that
sums up to the target integer . We prove that this problem cannot be
solved in time , for and any , assuming the Strong Exponential
Time Hypothesis (-SETH). We then use this result to exclude
-time algorithms for several
scheduling problems on jobs with maximum processing time , based
on -SETH. These include classical problems such as , the problem of minimizing the total weight of tardy jobs on a single
machine, and , the problem of minimizing the number of tardy
jobs on two identical parallel machines.Comment: 14 pages, ICALP'2
Treatment of Tricuspid Regurgitation With the FORMA Repair System
Background: Tricuspid regurgitation (TR) is common and undertreated as the risk of surgery is high in this patient population. Transcatheter devices offer treatment with a lower procedural risk. The FORMA Tricuspid Valve Therapy system (Edwards Lifesciences) will be reviewed here.Device Description: The system combines a spacer placed in the regurgitant orifice and a rail, over which the spacer is delivered, that is anchored to the endocardial surface of the RV. The spacer provides a surface for leaflet coaptation.Outcomes: Eighteen compassionate care patients and 29 patients included in the US EFS trial are reviewed. Patients were elderly (76 years) and high risk (Euroscore 2 was 9.0 and 8.1%, respectively). There were 2 procedural failures in both groups. Mortality at 30 days was 0% in the compassionate group and 7% in the EFS trial. TR was reduced in both groups; 2D/3D EROA 2.1 ± 1.8 to 1.1 ± 0.9 cm2 in the EFS trial and vena contracta width 12.1 ± 3.3 to 7.1 ± 2.2 mm. Symptomatic improvement was seen in both groups; the proportion of patients in NYHA class III/IV decreased from 84 to 28% at 30 days in the EFS group, and from 94 to 21% at 1 year, in the compassionate group.Conclusions: Reduction of TR with FORMA system is feasible and sustained. Despite residual TR post-procedure, the significant relative reduction in TR severity contributes to substantial clinical improvements in patients with a FORMA device in place
Faster Minimization of Tardy Processing Time on a Single Machine
This paper is concerned with the problem, the problem of minimizing the total processing time of tardy jobs on a single machine. This is not only a fundamental scheduling problem, but also a very important problem from a theoretical point of view as it generalizes the Subset Sum problem and is closely related to the 0/1-Knapsack problem. The problem is well-known to be NP-hard, but only in a weak sense, meaning it admits pseudo-polynomial time algorithms. The fastest known pseudo-polynomial time algorithm for the problem is the famous Lawler and Moore algorithm which runs in time, where is the total processing time of all jobs in the input. This algorithm has been developed in the late 60s, and has yet to be improved to date. In this paper we develop two new algorithms for , each improving on Lawler and Moore's algorithm in a different scenario. Both algorithms rely on basic primitive operations between sets of integers and vectors of integers for the speedup in their running times. The second algorithm relies on fast polynomial multiplication as its main engine, while for the first algorithm we define a new "skewed" version of -convolution which is interesting in its own right
Fairness in Repetitive Scheduling
Recent research found that fairness plays a key role in customer
satisfaction. Therefore, many manufacturing and services industries have become
aware of the need to treat customers fairly. Still, there is a huge lack of
models that enable industries to make operational decisions fairly, such as a
fair scheduling of the customers' jobs. Our main aim in this research is to
provide a unified framework to enable schedulers making fair decisions in
repetitive scheduling environments. For doing so, we consider a set of
repetitive scheduling problems involving a set of clients. In each out of
consecutive operational periods (e.g. days), each one of the customers
submits a job for processing by an operational system. The scheduler's aim is
to provide a schedule for each of the periods such that the quality of
service (QoS) received by each of the clients will meet a certain predefined
threshold. The QoS of a client may take several different forms, e.g., the
number of days that the customer receives its job later than a given due-date,
the number of times the customer receive his preferred time slot for service,
or the sum of waiting times for service. We analyze the single machine variant
of the problem for several different definitions of QoS, and classify the
complexity of the corresponding problems using the theories of classical and
parameterized complexity. We also study the price of fairness, i.e., the loss
in the system's efficiency that results from the need to provide fair
solutions
Scheduling Two Competing Agents When One Agent Has Significantly Fewer Jobs
We study a scheduling problem where two agents (each equipped with a private set of jobs) compete to perform their respective jobs on a common single machine. Each agent wants to keep the weighted sum of completion times of his jobs below a given (agent-dependent) bound. This problem is known to be NP-hard, even for quite restrictive settings of the problem parameters.
We consider parameterized versions of the problem where one of the agents has a small number of jobs (and where this small number constitutes the parameter). The problem becomes much more tangible in this case, and we present three positive algorithmic results for it. Our study is complemented by showing that the general problem is NP-complete even when one agent only has a single job
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