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

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

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 $2$-approximation algorithm for $P|G = block\ graph|C_{max}$ 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. $Q|G = k-block\ graph|C_{max}$. 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 $Q|G = 2-clique\ graph|C_{max}$, the result establishes the approximation status of $Q|G = k-block\ graph|C_{max}$. 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

Approximation Algorithms for Problems in Makespan Minimization on Unrelated Parallel Machines

A fundamental problem in scheduling is makespan minimization on unrelated parallel machines (R||Cmax). Let there be a set J of jobs and a set M of parallel machines, where every job Jj ∈ J has processing time or length pi,j ∈ ℚ+ on machine Mi ∈ M. The goal in R||Cmax is to schedule the jobs non-preemptively on the machines so as to minimize the length of the schedule, the makespan. A ρ-approximation algorithm produces in polynomial time a feasible solution such that its objective value is within a multiplicative factor ρ of the optimum, where ρ is called its approximation ratio. The best-known approximation algorithms for R||Cmax have approximation ratio 2, but there is no ρ-approximation algorithm with ρ \u3c 3/2 for R||Cmax unless P=NP. A longstanding open problem in approximation algorithms is to reconcile this hardness gap. We take a two-pronged approach to learn more about the hardness gap of R||Cmax: (1) find approximation algorithms for special cases of R||Cmax whose approximation ratios are tight (unless P=NP); (2) identify special cases of R||Cmax that have the same 3/2-hardness bound of R||Cmax, but where the approximation barrier of 2 can be broken. This thesis is divided into four parts. The first two parts investigate a special case of R||Cmax called the graph balancing problem when every job has one of two lengths and the machines may have one of two speeds. First, we present 3/2-approximation algorithms for the graph balancing problem with one speed and two job lengths. In the second part of this thesis we give an approximation algorithm for the graph balancing problem with two speeds and two job lengths with approximation ratio (√65+7)/8 ≈ 1.88278. In the third part of the thesis we present approximation algorithms and hardness of approximation results for two problems called R||Cmax with simple job-intersection structure and R||Cmax with bounded job assignments. We conclude this thesis by presenting algorithmic and computational complexity results for a generalization of R||Cmax where J is partitioned into sets called bags, and it must be that no two jobs belonging to the same bag are scheduled on the same machine