5 research outputs found
Approximation algorithms for job scheduling with block-type conflict graphs
The problem of scheduling jobs on parallel machines (identical, uniform, or
unrelated), under incompatibility relation modeled as a block graph, under the
makespan optimality criterion, is considered in this paper. No two jobs that
are in the relation (equivalently in the same block) may be scheduled on the
same machine in this model.
The presented model stems from a well-established line of research combining
scheduling theory with methods relevant to graph coloring. Recently, cluster
graphs and their extensions like block graphs were given additional attention.
We complement hardness results provided by other researchers for block graphs
by providing approximation algorithms. In particular, we provide a
-approximation algorithm for and a PTAS for the
case when the jobs are unit time in addition. In the case of uniform machines,
we analyze two cases. The first one is when the number of blocks is bounded,
i.e. . For this case, we provide a PTAS,
improving upon results presented by D. Page and R. Solis-Oba. The improvement
is two-fold: we allow richer graph structure, and we allow the number of
machine speeds to be part of the input. Due to strong NP-hardness of , the result establishes the approximation status of
. The PTAS might be of independent interest
because the problem is tightly related to the NUMERICAL k-DIMENSIONAL MATCHING
WITH TARGET SUMS problem. The second case that we analyze is when the number of
blocks is arbitrary, but the number of cut-vertices is bounded and jobs are of
unit time. In this case, we present an exact algorithm. In addition, we present
an FPTAS for graphs with bounded treewidth and a bounded number of unrelated
machines.Comment: 48 pages, 6 figures, 9 algorithm
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
Scheduling of Identical Jobs with Bipartite Incompatibility Graphs on Uniform Machines. Computational Experiments
Abstract. In the paper we consider the problem of scheduling of unit-length jobs on 3 or 4 uniform parallel machines to minimize schedule length or total completion time. We assume that jobs are subject to some kind of mutual exclusion constraints, modeled by a bipartite graph of bounded degree. The edges of the graph correspond to pairs of jobs that cannot be processed on the same machine. Although the problem is generally NP-hard, we show that under some conditions imposed on machine speeds and the structure of incompatibility graph our problem can be solved to optimality in polynomial time. Theoretical considerations are accompanied by computer experiments with some particular model of scheduling
Scheduling with complete multipartite incompatibility graph on parallel machines
In this paper we consider a problem of job scheduling on parallel machines with a presence of incompatibilities between jobs. The incompatibility relation can be modeled as a complete multipartite graph in which each edge denotes a pair of jobs that cannot be scheduled on the same machine. We provide several results concerning schedules, optimal or approximate with respect to the two most popular criteria of optimality: Cmax (makespan) and ∑Cj (total completion time). We consider a variety of machine types in our paper: identical, uniform, and unrelated. Our results consist of delimitation of the easy (polynomial) and NP-hard problems within these constraints. We also provide algorithms, either polynomial exact algorithms for the easier problems, or algorithms with a guaranteed constant worst-case approximation ratio. In particular, we fill the gap on research for the problem of finding a schedule with the smallest ∑Cj on uniform machines. We address this problem by developing a linear programming relaxation technique with an appropriate rounding, which to our knowledge is a novelty for this criterion in the considered setting