1,003 research outputs found

### Relativity of a Free Will Concept Depending on Both Conscious Indeterminism and Unconscious Determinism

Free will is difficult to classify with respect to determinism or indeterminism, and its phenomenology in consciousness often shows both aspects. Initially, it is felt as unlimited and indeterminate will power, with the potentiality of multiple choices. Thereafter, reductive deliberation is led by determinism to the final decision, which realises only one of the potential choices. The reductive deliberation phase tries to find out the best alternative and simultaneously satisfying vague motivations, contextual conditions and personal preferences. The essential sense of free will is the introduction of personal preferences, which allows a higher diversity of reactions to vague motivations. With an oversimplified model of determinism as a chain of events, incompatibilists define â€śfreeâ€ť as â€śundeterminedâ€ť so that determinism becomes incompatible with any free choice between alternatives. In consciousness, free will requires a more complex model of network determinism as well as the consideration of unconsciousness as a causal factor. When â€śfreeâ€ť defined as â€śundeterminedâ€ť is applied to the context of consciousness, it should be reinterpreted as â€śunconscious of being determinedâ€ť or not aware of underlying determinism. Lacking information on determinism generates a feeling of â€śfreeâ€ť in consciousness and, therefore, gives the impression of indeterminism. Lacking information may be induced by an uncertain future without determined eventsâ€”an unconscious past with biological reactions suddenly emerging from the unconsciousness or an unknown present unable to distinguish determinism of complex events. Therefore, at the level of human consciousness, the experience of free will is associated with apparent indeterminism although it is based on unconscious determinism. The concepts of compatibilism and incompatibilism are only two different aspects of the same phenomenon and correspond to consciousness and unconsciousness. Nevertheless, they can be considered together with a free will concept based on relativity depending on two different reference framesâ€”the first personâ€™s experience frame or the Laplaceâ€™s demon frame with knowledge on every molecule of the universe. Only relativity of the free will concept avoids the contradiction between â€śfreeâ€ť and â€śunfreeâ€ť for the same phenomenon and could be a compromise for considering compatibilism and incompatibilism equally

### Closing the Gap for Pseudo-Polynomial Strip Packing

Two-dimensional packing problems are a fundamental class of optimization problems and Strip Packing is one of the most natural and famous among them. Indeed it can be defined in just one sentence: Given a set of rectangular axis parallel items and a strip with bounded width and infinite height, the objective is to find a packing of the items into the strip minimizing the packing height. We speak of pseudo-polynomial Strip Packing if we consider algorithms with pseudo-polynomial running time with respect to the width of the strip. It is known that there is no pseudo-polynomial time algorithm for Strip Packing with a ratio better than 5/4 unless P = NP. The best algorithm so far has a ratio of 4/3 + epsilon. In this paper, we close the gap between inapproximability result and currently known algorithms by presenting an algorithm with approximation ratio 5/4 + epsilon. The algorithm relies on a new structural result which is the main accomplishment of this paper. It states that each optimal solution can be transformed with bounded loss in the objective such that it has one of a polynomial number of different forms thus making the problem tractable by standard techniques, i.e., dynamic programming. To show the conceptual strength of the approach, we extend our result to other problems as well, e.g., Strip Packing with 90 degree rotations and Contiguous Moldable Task Scheduling, and present algorithms with approximation ratio 5/4 + epsilon for these problems as well

### Scheduling Monotone Moldable Jobs in Linear Time

A moldable job is a job that can be executed on an arbitrary number of
processors, and whose processing time depends on the number of processors
allotted to it. A moldable job is monotone if its work doesn't decrease for an
increasing number of allotted processors. We consider the problem of scheduling
monotone moldable jobs to minimize the makespan.
We argue that for certain compact input encodings a polynomial algorithm has
a running time polynomial in n and log(m), where n is the number of jobs and m
is the number of machines. We describe how monotony of jobs can be used to
counteract the increased problem complexity that arises from compact encodings,
and give tight bounds on the approximability of the problem with compact
encoding: it is NP-hard to solve optimally, but admits a PTAS.
The main focus of this work are efficient approximation algorithms. We
describe different techniques to exploit the monotony of the jobs for better
running times, and present a (3/2+{\epsilon})-approximate algorithm whose
running time is polynomial in log(m) and 1/{\epsilon}, and only linear in the
number n of jobs

### On Integer Programming, Discrepancy, and Convolution

Integer programs with a constant number of constraints are solvable in
pseudo-polynomial time. We give a new algorithm with a better pseudo-polynomial
running time than previous results. Moreover, we establish a strong connection
to the problem (min, +)-convolution. (min, +)-convolution has a trivial
quadratic time algorithm and it has been conjectured that this cannot be
improved significantly. We show that further improvements to our
pseudo-polynomial algorithm for any fixed number of constraints are equivalent
to improvements for (min, +)-convolution. This is a strong evidence that our
algorithm's running time is the best possible. We also present a faster
specialized algorithm for testing feasibility of an integer program with few
constraints and for this we also give a tight lower bound, which is based on
the SETH.Comment: A preliminary version appeared in the proceedings of ITCS 201

### Inapproximability Results for Scheduling with Interval and Resource Restrictions

In the restricted assignment problem, the input consists of a set of machines and a set of jobs each with a processing time and a subset of eligible machines. The goal is to find an assignment of the jobs to the machines minimizing the makespan, that is, the maximum summed up processing time any machine receives. Herein, jobs should only be assigned to those machines on which they are eligible. It is well-known that there is no polynomial time approximation algorithm with an approximation guarantee of less than 1.5 for the restricted assignment problem unless P=NP. In this work, we show hardness results for variants of the restricted assignment problem with particular types of restrictions.
For the case of interval restrictions - where the machines can be totally ordered such that jobs are eligible on consecutive machines - we show that there is no polynomial time approximation scheme (PTAS) unless P=NP. The question of whether a PTAS for this variant exists was stated as an open problem before, and PTAS results for special cases of this variant are known.
Furthermore, we consider a variant with resource restriction where the sets of eligible machines are of the following form: There is a fixed number of (renewable) resources, each machine has a capacity, and each job a demand for each resource. A job is eligible on a machine if its demand is at most as big as the capacity of the machine for each resource. For one resource, this problem has been intensively studied under several different names and is known to admit a PTAS, and for two resources the variant with interval restrictions is contained as a special case. Moreover, the version with multiple resources is closely related to makespan minimization on parallel machines with a low rank processing time matrix. We show that there is no polynomial time approximation algorithm with a rate smaller than 48/47 ? 1.02 or 1.5 for scheduling with resource restrictions with 2 or 4 resources, respectively, unless P=NP. All our results can be extended to the so called Santa Claus variants of the problems where the goal is to maximize the minimal processing time any machine receives

### An EPTAS for machine scheduling with bag-constraints

Machine scheduling is a fundamental optimization problem in computer science.
The task of scheduling a set of jobs on a given number of machines and
minimizing the makespan is well studied and among other results, we know that
EPTAS's for machine scheduling on identical machines exist. Das and Wiese
initiated the research on a generalization of makespan minimization, that
includes so called bag-constraints. In this variation of machine scheduling the
given set of jobs is partitioned into subsets, so called bags. Given this
partition a schedule is only considered feasible when on any machine there is
at most one job from each bag.
Das and Wiese showed that this variant of machine scheduling admits a PTAS.
We will improve on this result by giving the first EPTAS for the machine
scheduling problem with bag-constraints. We achieve this result by using new
insights on this problem and restrictions given by the bag-constraints. We show
that, to gain an approximate solution, we can relax the bag-constraints and
ignore some of the restrictions. Our EPTAS uses a new instance transformation
that will allow us to schedule large and small jobs independently of each other
for a majority of bags. We also show that it is sufficient to respect the
bag-constraint only among a constant number of bags, when scheduling large
jobs. With these observations our algorithm will allow for some conflicts when
computing a schedule and we show how to repair the schedule in polynomial-time
by swapping certain jobs around

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