6 research outputs found
Fuzzy Simultaneous Congruences
We introduce a very natural generalization of the well-known problem of
simultaneous congruences. Instead of searching for a positive integer that
is specified by fixed remainders modulo integer divisors we
consider remainder intervals such that is feasible if and
only if is congruent to modulo for some remainder in
interval for all .
This problem is a special case of a 2-stage integer program with only two
variables per constraint which is is closely related to directed Diophantine
approximation as well as the mixing set problem. We give a hardness result
showing that the problem is NP-hard in general.
By investigating the case of harmonic divisors, i.e. is an
integer for all , which was heavily studied for the mixing set problem as
well, we also answer a recent algorithmic question from the field of real-time
systems. We present an algorithm to decide the feasibility of an instance in
time and we show that if it exists even the smallest
feasible solution can be computed in strongly polynomial time
Scheduling with Many Shared Resources
Consider the many shared resource scheduling problem where jobs have to be
scheduled on identical parallel machines with the goal of minimizing the
makespan. However, each job needs exactly one additional shared resource in
order to be executed and hence prevents the execution of jobs that need the
same resource while being processed. Previously a -approximation
was the best known result for this problem. Furthermore, a -approximation
for the case with only two machines was known as well as a PTAS for the case
with a constant number of machines. We present a simple and fast
5/3-approximation and a much more involved but still reasonable
1.5-approximation. Furthermore, we provide a PTAS for the case with only a
constant number of machines, which is arguably simpler and faster than the
previously known one, as well as a PTAS with resource augmentation for the
general case. The approximation schemes make use of the N-fold integer
programming machinery, which has found more and more applications in the field
of scheduling recently. It is plausible that the latter results can be improved
and extended to more general cases. Lastly, we give a
inapproximability result for the natural problem extension where each job may
need up to a constant number (in particular ) of different resources
Load Balancing: The Long Road from Theory to Practice
There is a long history of approximation schemes for the problem of
scheduling jobs on identical machines to minimize the makespan. Such a scheme
grants a -approximation solution for every , but
the running time grows exponentially in . For a long time, these
schemes seemed like a purely theoretical concept. Even solving instances for
moderate values of seemed completely illusional. In an effort to
bridge theory and practice, we refine recent ILP techniques to develop the
fastest known approximation scheme for this problem. An implementation of this
algorithm reaches values of lower than within a
reasonable timespan. This is the approximation guarantee of MULTIFIT, which, to
the best of our knowledge, has the best proven guarantee of any non-scheme
algorithm
Load Balancing: The Long Road from Theory to Practice
There is a long history of approximation schemes for the problem of scheduling jobs on identical machines to minimize the makespan. Such a scheme grants a (1 + ε)-approximation solution for every ε > 0, but the running time grows exponentially in 1/ε. For a long time, these schemes seemed like a purely theoretical concept. Even solving instances for moderate values of ε seemed completely illusional. In an effort to bridge theory and practice, we refine recent ILP techniques to develop the fastest known approximation scheme for this problem. An implementation of this algorithm reaches values of ε lower than 2/11 ≈ 18.2% within a reasonable timespan. This is the approximation guarantee of MULTIFIT, which, to the best of our knowledge, has the best proven guarantee of any non-scheme algorithm