The generalized work function algorithm is competitive for the generalized 2-server problem

The generalized 2-server problem is an online optimization problem where a sequence of requests has to be served at minimal cost. Requests arrive one by one and need to be served instantly by at least one of two servers. We consider the general model where the cost function of the two servers may be different. Formally, each server moves in its own metric space and a request consists of one point in each metric space. It is served by moving one of the two servers to its request point. Requests have to be served without knowledge of the future requests. The objective is to minimize the total traveled distance. The special case where both servers move on the real line is known as the CNN-problem. We show that the generalized work function algorithm is constant competitive for the generalized 2-server problem

Split Scheduling with Uniform Setup Times

We study a scheduling problem in which jobs may be split into parts, where the parts of a split job may be processed simultaneously on more than one machine. Each part of a job requires a setup time, however, on the machine where the job part is processed. During setup a machine cannot process or set up any other job. We concentrate on the basic case in which setup times are job-, machine-, and sequence-independent. Problems of this kind were encountered when modelling practical problems in planning disaster relief operations. Our main algorithmic result is a polynomial-time algorithm for minimising total completion time on two parallel identical machines. We argue why the same problem with three machines is not an easy extension of the two-machine case, leaving the complexity of this case as a tantalising open problem. We give a constant-factor approximation algorithm for the general case with any number of machines and a polynomial-time approximation scheme for a fixed number of machines. For the version with objective minimising weighted total completion time we prove NP-hardness. Finally, we conclude with an overview of the state of the art for other split scheduling problems with job-, machine-, and sequence-independent setup times

On the value of preemption in scheduling

It is well known that on-line preemptive scheduling algorithms can achieve efficient performance, A classic example is the Shortest Remaining Processing Time (SRPT) algorithm which is optimal for flow time scheduling, assuming preemption is costless. In real systems, however, preemption has significant overhead. In this paper we suggest a new model where preemption is costly. This introduces new considerations for preemptive scheduling algorithms and inherently calls for new scheduling strategies. We present a simple on-line algorithm and present lower bounds for on-line as well as efficient off-line algorithms which show that our algorithm performs close to optimal

On the value of preemption in scheduling

It is well known that on-line preemptive scheduling algorithms can achieve efficient performance, A classic example is the Shortest Remaining Processing Time (SRPT) algorithm which is optimal for flow time scheduling, assuming preemption is costless. In real systems, however, preemption has significant overhead. In this paper we suggest a new model where preemption is costly. This introduces new considerations for preemptive scheduling algorithms and inherently calls for new scheduling strategies. We present a simple on-line algorithm and present lower bounds for on-line as well as efficient off-line algorithms which show that our algorithm performs close to optimal. © Springer-Verlag Berlin Heidelberg 2006

How to sell a graph: guidelines for graph retailers

We consider a profit maximization problem where we are asked to price a set of $m$ items that are to be assigned to a set of $n$ customers. The items can be represented as the edges of an undirected (multi)graph $G$, where an edge multiplicity larger than one corresponds to multiple copies of the same item. Each customer is interested in purchasing a bundle of edges of $G$, and we assume that each bundle forms a simple path in $G$. Each customer has a known budget for her respective bundle, and is interested only in that particular bundle. The goal is to determine item prices and a feasible assignment of items to customers in order to maximize the total profit. When the underlying graph $G$ is a path, we derive a fully polynomial time approximation scheme, complementing a recent NP-hardness result. If the underlying graph is a tree, and edge multiplicities are one, we show that the problem is polynomially solvable, contrasting its APX-hardness for the case of unlimited availability of items. However, if the underlying graph is a grid, and edge multiplicities are one, we show that it is even NP-complete to approximate the maximum profit to within a factor $n^{1-\varepsilon}$

The Sorting Buffer Problem is NP-hard

We consider the offline sorting buffer problem. The input is a sequence of items of different types. All items must be processed one by one by a server. The server is equipped with a random-access buffer of limited capacity which can be used to rearrange items. The problem is to design a scheduling strategy that decides upon the order in which items from the buffer are sent to the server. Each type change incurs unit cost, and thus, the cost minimizing objective is to minimize the total number of type changes for serving the entire sequence. This problem is motivated by various applications in manufacturing processes and computer science, and it has attracted significant attention in the last few years. The main focus has been on online competitive algorithms. Surprisingly little is known on the basic offline problem. In this paper, we show that the sorting buffer problem with uniform cost is NP-hard and, thus, close one of the most fundamental questions for the offline problem. On the positive side, we give an O(1)-approximation algorithm when the scheduler is given a buffer only slightly larger than double the original size. We also give a dynamic programming algorithm for the special case of buffer size two that solves the problem exactly in linear time, improving on the standard DP which runs in cubic time

Scheduling over scenarios on two machines

We consider scheduling problems over scenarios where the goal is to find a single assignment of the jobs to the machines which performs well over all possible scenarios. Each scenario is a subset of jobs that must be executed in that scenario and all scenarios are given explicitly. The two objectives that we consider are minimizing the maximum makespan over all scenarios and minimizing the sum of the makespans of all scenarios. For both versions, we give several approximation algorithms and lower bounds on their approximability. With this research into optimization problems over scenarios, we have opened a new and rich field of interesting problems. © 2014 Springer International Publishing Switzerland