Additive Approximation Schemes for Load Balancing Problems

We formalize the concept of additive approximation schemes and apply it to load balancing problems on identical machines. Additive approximation schemes compute a solution with an absolute error in the objective of at most ? h for some suitable parameter h and any given ? > 0. We consider the problem of assigning jobs to identical machines with respect to common load balancing objectives like makespan minimization, the Santa Claus problem (on identical machines), and the envy-minimizing Santa Claus problem. For these settings we present additive approximation schemes for h = p_{max}, the maximum processing time of the jobs. Our technical contribution is two-fold. First, we introduce a new relaxation based on integrally assigning slots to machines and fractionally assigning jobs to the slots. We refer to this relaxation as the slot-MILP. While it has a linear number of integral variables, we identify structural properties of (near-)optimal solutions, which allow us to compute those in polynomial time. The second technical contribution is a local-search algorithm which rounds any given solution to the slot-MILP, introducing an additive error on the machine loads of at most ?? p_{max}

Scheduling with Machine Conflicts

We study the scheduling problem of makespan minimization while taking machine conflicts into account. Machine conflicts arise in various settings, e.g., shared resources for pre- and post-processing of tasks or spatial restrictions. In this context, each job has a blocking time before and after its processing time, i.e., three parameters. We seek for conflict-free schedules in which the blocking times of no two jobs intersect on conflicting machines. Given a set of jobs, a set of machines, and a graph representing machine conflicts, the problem SchedulingWithMachineConflicts (SMC), asks for a conflict-free schedule of minimum makespan. We show that, unless $\textrm{P}=\textrm{NP}$, SMC on $m$ machines does not allow for a $\mathcal{O}(m^{1-\varepsilon})$-approximation algorithm for any $\varepsilon>0$, even in the case of identical jobs and every choice of fixed positive parameters, including the unit case. Complementary, we provide approximation algorithms when a suitable collection of independent sets is given. Finally, we present polynomial time algorithms to solve the problem for the case of unit jobs on special graph classes. Most prominently, we solve it for bipartite graphs by using structural insights for conflict graphs of star forests.Comment: 20 pages, 8 figure

A (3+ϵ)(3+\epsilon)-approximation algorithm for the minimum sum of radii problem with outliers and extensions for generalized lower bounds

For a given set of points in a metric space and an integer $k$, we seek to partition the given points into $k$ clusters. For each computed cluster, one typically defines one point as the center of the cluster. A natural objective is to minimize the sum of the cluster center's radii, where we assign the smallest radius $r$ to each center such that each point in the cluster is at a distance of at most $r$ from the center. The best-known polynomial time approximation ratio for this problem is $3.389$. In the setting with outliers, i.e., we are given an integer $m$ and allow up to $m$ points that are not in any cluster, the best-known approximation factor is $12.365$. In this paper, we improve both approximation ratios to $3+\epsilon$. Our algorithms are primal-dual algorithms that use fundamentally new ideas to compute solutions and to guarantee the claimed approximation ratios. For example, we replace the classical binary search to find the best value of a Lagrangian multiplier $\lambda$ by a primal-dual routine in which $\lambda$ is a variable that is raised. Also, we show that for each connected component due to almost tight dual constraints, we can find one single cluster that covers all its points and we bound its cost via a new primal-dual analysis. We remark that our approximation factor of $3+\epsilon$ is a natural limit for the known approaches in the literature. Then, we extend our results to the setting of lower bounds. There are algorithms known for the case that for each point $i$ there is a lower bound $L_{i}$, stating that we need to assign at least $L_{i}$ clients to $i$ if $i$ is a cluster center. For this setting, there is a $3.83$ approximation if outliers are not allowed and a ${12.365}$-approximation with outliers. We improve both ratios to $3.5 + \epsilon$ and, at the same time, generalize the type of allowed lower bounds

Hiking in the scheduling landscape:exact and approximation algorithms for parallel machines

This thesis investigates different optimization problems in the field of scheduling. Scheduling problems model situations in which limited resources have to be assigned to tasks over time as to minimize costs, maximize profits, balance workloads among resources or improve the efficiency of the usage of resources. The goal is to develop algorithms, which can find optimal solutions in an efficient amount of time. However, many scheduling problems turn out to be computationally difficult such that we cannot hope to accomplish this goal. To overcome this, we consider approximation algorithms, which offer a trade-off by computing solutions provably close to an optimal solution in an efficient amount of time. In this thesis both exact and approximation algorithms are studied for a variety of scheduling problems. To achieve this, this research investigates the structure and properties of optimal solutions to find solutions that are provably close

Performance analysis of fixed assignment policies for stochastic online scheduling on uniform parallel machines

In stochastic online scheduling problems, a common class of policies is the class of fixed assignment policies. These policies first assign jobs to machines and then apply single machine scheduling policies for each machine separately. We consider a stochastic online scheduling problem for which the goal is to minimize total weighted expected completion time on uniform parallel machines. To solve the problem, we adapt policies introduced for the identical and unrelated parallel machine environments. We show that, with the help of lower bounds specific for the uniform machine environment, we can tighten the performance guarantees that are implied by the results for the unrelated machine environment for the special case of two machine speeds. Furthermore, in the Online-List model we show that a greedy assignment policy is asymptotically optimal. Finally, we construct a computational study to assess the performance of the policies in practice

Vessel velocity decisions in inland waterway transportation under uncertainty

Recent studies have concentrated on environmental and economic impacts of ships. In this regard, fuel and CO2 emission is considered as one of the important factors for such impacts. In particular, the sailing speed of the vessels affects the fuel consumption and therefore the emission directly. In this study, we consider a speed optimization problem in inland waterway, which is characterized by stochastic waiting times at the lock caused by uncertainty in lock processing time estimations of other vessels. The objective is to minimize fuel consumption of an approaching ship, such that it traverses the river segment in a set deadline. We introduce a mathematical model for this problem and evaluate the effectiveness and attractiveness of two solution approaches: an optimal solution and a simple heuristic. This creates intuitive guidelines for skippers based on information provision to select an appropriate speed decision approach to minimize the total expected fuel consumption and CO2 emission of inland waterway transportation

Additive approximation and approximation schemes for load balancing

Many applications in discrete optimization lead to hard problems. Under common assumption, it is impossible to find an algorithm that (1) is efficient, (2) finds an optimal solution on (3) every instance. At least one of these requirements needs to be sacrificed to cope with these problems. In the area of approximation algorithms, the goal is to design algorithms that efficiently find provably good solutions. Typically, for approximation algorithms, provably good implies that we bound the approximation ratio of the value of the solution to the optimal value. One important reason for studying approximation algorithms is that often even on simplified problems, they give us insights in how to design heuristics for the real problem that needs to be solved. Furthermore, having a mathematical proof for an approximation guarantee often results in a deeper understanding of the structure of the underlying problem. Unfortunately, in some cases finding a guarantee on the approximation ratio is impossible, e.g., when the optimal solution value is 0. Or the approximation guarantee is overly pessimistic, e.g., Graham’s (1966) seminal List Scheduling algorithm for makespanscheduling is guaranteed to find a solution with value at most twice the optimal value, but when processing times are small List Scheduling performs much better. To overcome these issues, we consider the concept of additive approximation algorithms. Instead of bounding the ratio, in additive approximation we bound the absolute difference between the value of the solution of the approximation algorithm and the optimal solution value. We apply the concept of additive approximation and additive approximation schemes, that can get arbitrarily close to an optimal solution, for several load balancing problems.<br/

Vessel velocity decisions in inland waterway transportation under uncertainty

Recent studies have concentrated on environmental and economic impacts of ships. In this regard, fuel and CO2 emission is considered as one of the important factors for such impacts. In particular, the sailing speed of the vessels affects the fuel consumption and therefore the emission directly. In this study, we consider a speed optimization problem in inland waterway, which is characterized by stochastic waiting times at the lock caused by uncertainty in lock processing time estimations of other vessels. The objective is to minimize fuel consumption of an approaching ship, such that it traverses the river segment in a set deadline. We introduce a mathematical model for this problem and evaluate the effectiveness and attractiveness of two solution approaches: an optimal solution and a simple heuristic. This creates intuitive guidelines for skippers based on information provision to select an appropriate speed decision approach to minimize the total expected fuel consumption and CO2 emission of inland waterway transportation

Additive Approximation Schemes for Load Balancing Problems

We formalize the concept of additive approximation schemes and apply it to load balancing problems on identical machines. Additive approximation schemes compute a solution with an absolute error in the objective of at most ε h for some suitable parameter h and any given ε > 0. We consider the problem of assigning jobs to identical machines with respect to common load balancing objectives like makespan minimization, the Santa Claus problem (on identical machines), and the envy-minimizing Santa Claus problem. For these settings we present additive approximation schemes for h = p_{max}, the maximum processing time of the jobs. Our technical contribution is two-fold. First, we introduce a new relaxation based on integrally assigning slots to machines and fractionally assigning jobs to the slots. We refer to this relaxation as the slot-MILP. While it has a linear number of integral variables, we identify structural properties of (near-)optimal solutions, which allow us to compute those in polynomial time. The second technical contribution is a local-search algorithm which rounds any given solution to the slot-MILP, introducing an additive error on the machine loads of at most ε⋅ p_{max}