5 research outputs found
Parameterized and approximation results for scheduling with a low rank processing time matrix
We study approximation and parameterized algorithms for R||C_max, focusing on the problem when the rank of the matrix formed by job processing times is small. Bhaskara et al. initiated the study of approximation algorithms with respect to the rank, showing that R||C_max admits a QPTAS (Quasi-polynomial time approximation scheme) when the rank is 2, and becomes APX-hard when the rank is 4.
We continue this line of research. We prove that R||C_max is APX-hard even if the rank is 3, resolving an open problem. We then show that R||C_max is FPT parameterized by the rank and the largest job processing time p_max. This generalizes the parameterized results on P||C_max and R||C_max with few different types of machines. We also provide nearly tight lower bounds under Exponential Time Hypothesis which suggests that the running time of the FPT algorithm is unlikely to be improved significantly
Parameterized complexity of machine scheduling: 15 open problems
Machine scheduling problems are a long-time key domain of algorithms and
complexity research. A novel approach to machine scheduling problems are
fixed-parameter algorithms. To stimulate this thriving research direction, we
propose 15 open questions in this area whose resolution we expect to lead to
the discovery of new approaches and techniques both in scheduling and
parameterized complexity theory.Comment: Version accepted to Computers & Operations Researc
An EPTAS for Scheduling on Unrelated Machines of Few Different Types
In the classical problem of scheduling on unrelated parallel machines, a set
of jobs has to be assigned to a set of machines. The jobs have a processing
time depending on the machine and the goal is to minimize the makespan, that is
the maximum machine load. It is well known that this problem is NP-hard and
does not allow polynomial time approximation algorithms with approximation
guarantees smaller than unless PNP. We consider the case that there
are only a constant number of machine types. Two machines have the same
type if all jobs have the same processing time for them. This variant of the
problem is strongly NP-hard already for . We present an efficient
polynomial time approximation scheme (EPTAS) for the problem, that is, for any
an assignment with makespan of length at most
times the optimum can be found in polynomial time in the
input length and the exponent is independent of . In particular
we achieve a running time of , where
denotes the input length. Furthermore, we study three other problem
variants and present an EPTAS for each of them: The Santa Claus problem, where
the minimum machine load has to be maximized; the case of scheduling on
unrelated parallel machines with a constant number of uniform types, where
machines of the same type behave like uniformly related machines; and the
multidimensional vector scheduling variant of the problem where both the
dimension and the number of machine types are constant. For the Santa Claus
problem we achieve the same running time. The results are achieved, using mixed
integer linear programming and rounding techniques
Total Completion Time Minimization for Scheduling with Incompatibility Cliques
This paper considers parallel machine scheduling with incompatibilities
between jobs. The jobs form a graph and no two jobs connected by an edge are
allowed to be assigned to the same machine. In particular, we study the case
where the graph is a collection of disjoint cliques. Scheduling with
incompatibilities between jobs represents a well-established line of research
in scheduling theory and the case of disjoint cliques has received increasing
attention in recent years. While the research up to this point has been focused
on the makespan objective, we broaden the scope and study the classical total
completion time criterion. In the setting without incompatibilities, this
objective is well known to admit polynomial time algorithms even for unrelated
machines via matching techniques. We show that the introduction of
incompatibility cliques results in a richer, more interesting picture.
Scheduling on identical machines remains solvable in polynomial time, while
scheduling on unrelated machines becomes APX-hard. Furthermore, we study the
problem under the paradigm of fixed-parameter tractable algorithms (FPT). In
particular, we consider a problem variant with assignment restrictions for the
cliques rather than the jobs. We prove that it is NP-hard and can be solved in
FPT time with respect to the number of cliques. Moreover, we show that the
problem on unrelated machines can be solved in FPT time for reasonable
parameters, e.g., the parameter pair: number of machines and maximum processing
time. The latter result is a natural extension of known results for the case
without incompatibilities and can even be extended to the case of total
weighted completion time. All of the FPT results make use of n-fold Integer
Programs that recently have received great attention by proving their
usefulness for scheduling problems