Ariella Voloshin

Optimizing busy time on parallel machines

We consider the following fundamental parallel machines scheduling problem in which the input consists of n jobs to be scheduled on a set of identical machines of bounded capacity g, which is the maximal number of jobs that can be processed simultaneously by a single machine. Each job is associated with a time interval during which it should be processed from start to end (and in one of our extensions it has to be scheduled also in a continuous number of days; this corresponds to a two-dimensional variant of the problem). We consider two versions of the problem. In the scheduling minimization version the goal is to minimize the total busy time of machines used to schedule all jobs. In the resource allocation maximization version the goal is to maximize the number of jobs that can be scheduled for processing under a budget constraint given in terms of busy time. This is the first study of the maximization version of the problem. The minimization problem is known to be NP-Hard, thus the maximization problem is also NP-Hard. We consider various special cases, identify cases where an optimal solution can be computed in polynomial time, and mainly provide constant factor approximation algorithms for both minimization and maximization problems. Some of our results improve upon the best known results for this job scheduling problem. Our study has applications in energy-aware scheduling, cloud computing, switching cost optimization as well as wavelength assignments in optical networks.

Optimizing Busy Time on Parallel Machines

We consider the following fundamental scheduling problem in which the input consists of n jobs to be scheduled on a set of identical machines of bounded capacity g (which is the maximal number of jobs that can be processed simultaneously by a single machine). Each job is associated with a start time and a completion time, it is supposed to be processed from the start time to the completion time (and in one of our extensions it has to be scheduled also in a continuous number of days, this corresponds to a two-dimensional version of the problem). We consider two versions of the problem. In the scheduling minimization version the goal is to minimize the total busy time of machines used to schedule all jobs. In the resource allocation maximization version the goal is to maximize the number of jobs that are scheduled for processing under a budget constraint given in terms of busy time. This is the first study of the maximization version of the problem. The minimization problem is known to be NP-Hard, thus the maximization problem is also NP-Hard. We consider various special cases, identify cases where an optimal solution can be computed in polynomial time, and mainly provide constant factor approximation algorithms for both minimization and maximization problems. Some of our results improve upon the best known results for this job scheduling problem. Our study has applications in power consumption, cloud computing and optimizing switching cost of optical networks.

Online optimization of busy time on parallel machines

We consider the following online scheduling problem in which the input consists of n jobs to be scheduled on identical machines of bounded capacity g (the maximum number of jobs that can be processed simultaneously on a single machine). Each job is associated with a release time and a completion time between which it is supposed to be processed. When a job is released, the online algorithm has to make decision without changing it afterwards. We consider two versions of the problem. In the minimization version, the goal is to minimize the total busy time of machines used to schedule all jobs. In the resource allocation maximization version, the goal is to maximize the number of jobs that are scheduled under a budget constraint given in terms of busy time. This is the first study on online algorithms for these problems. We show a rather large lower bound on the competitive ratio for general instances. This motivates us to consider special families of input instances for which we show constant competitive algorithms. Our study has applications in power aware scheduling, cloud computing and optimizing switching cost of optical networks.

Optimizing Bandwidth Allocation in Flex-Grid Optical Networks with Application to Scheduling

All-optical networks have been largely investigated due to their high data transmission rates. In the traditional Wavelength-Division Multiplexing (WDM) technology, the spectrum of light that can be transmitted through the optical fiber has been divided into frequency intervals of fixed width, with a gap of unused frequencies between them. Recently, an alternative emerging architecture was suggested which moves away from the rigid Dense WDM (DWDM) model towards a flexible model, where usable frequency intervals are of variable width (even within the same link). Each light path has to be assigned a frequency interval (sub-spectrum), which remains fixed through all of the links it traverses. Two different light paths using the same link must be assigned disjoint sub-spectra. This technology is termed flex-grid (or, flex-spectrum), as opposed to fixed-grid (or, fixed-spectrum) current technology. In this work we study a problem of optimal bandwidth allocation arising in the flex-grid technology. In this setting, each light path has a lower and upper bound on the width of its frequency interval, as well as an associated profit, and we want to find a bandwidth assignment that maximizes the total profit. This problem is known to be NP-Complete. We observe that, in fact, the problem is inapproximable within any constant ratio even on a path network. We further derive NP-hardness results and present approximation algorithms for several special cases of the path and ring networks, which are of practical interest. Finally, while in general our problem is hard to approximate, we show that an optimal solution can be obtained by allowing resource augmentation. Our study has applications also in real time scheduling.

Flexible Bandwidth Assignment with Application to Optical Networks

We introduce two scheduling problems, the flexible bandwidth allocation problem (FBAP) and the flexible storage allocation problem (FSAP). In both problems, we have an available resource, and a set of requests, each consists of a minimum and a maximum resource requirement, for the duration of its execution, as well as a profit accrued per allocated unit of the resource. In FBAP the goal is to assign the available resource to a feasible subset of requests, such that the total profit is maximized, while in FSAP we also require that each satisfied request is given a contiguous portion of the resource. Our problems generalize the classic bandwidth allocation problem (BAP) and storage allocation problem (SAP) and are therefore NP-Hard.

On the complexity of constructing minimum changeover cost arborescences

The reload cost concept refers to the cost that occurs at a vertex along a path on an edge-colored graph when it traverses an internal vertex between two edges of different colors. This cost depends only on the colors of the traversed edges. Reload costs arise in various applications such as transportation networks, energy distribution networks, and telecommunications. Previous work on reload costs focuses on two problems of finding a spanning tree with minimum cost with respect to two different cost measures. In both problems the cost is associated with a set of paths from a given vertex r to all the leaves of the constructed tree. The first cost measure is the sum of the reload costs of all paths from r to the leaves. The second cost measure is the changeover cost, in which the cost of traversing a vertex by using two specific incident edges is paid only once regardless of the number of paths traversing it. The first problem is inapproximable within any polynomial time computable function of the input size [1], and the second problem is inapproximable within n1−ϵ for any ϵ>0 [2]. In this paper we show that the first hardness result holds also for the second problem. Given this strong inapproximability result, we study the complexity and approximability properties of numerous special cases of this second problem. We mainly focus on bounded costs, and consider both directed and undirected graphs, bounded and unbounded number of colors, and both bounded and unbounded degree graphs. We also present polynomial time exact algorithms and an approximation algorithm for some special case. To the best of our knowledge, these are the first algorithms with a provable performance guarantee for the problem. Moreover, our approximation algorithm shows a tight bound on the approximability of the problem for a specific family of instances.

Flexible Cell Selection in Cellular Networks

We introduce the problem of Flexible Scheduling on Related Machines with Assignment Restrictions (FSRM). In this problem the input consists of a set of machines and a set of jobs. Each machine has a finite capacity, and each job has a resource requirement interval, a profit per allocated unit of resource, and a set of machines that can potentially supply the requirement. A feasible solution is an allocation of machine resources to jobs such that: (i) a machine resource can be allocated to a job only if it is a potential supplier of this job, (ii) the amount of machine resources allocated by a machine is bounded by its capacity, and (iii) the amount of resources that are allocated to a job is either in its requirement interval or zero. Notice that a job can be serviced by multiple machines. The goal is to find a feasible allocation that maximizes the overall profit. We focus on r-FSRM in which the required resource of a job is at most an r-fraction of (or r times) the capacity of each potential machine. FSRM is motivated by resource allocation problems arising in cellular networks and in cloud computing. Specifically, FSRM models the problem of assigning clients to base stations in 4G cellular networks. We present a 2-approximation algorithm for 1-FSRM and a \(\frac{1}{1-r}\)-approximation algorithm for r-FSRM, for any \(r \in (0,1)\). Both are based on the local ratio technique and on maximum flow computations. We also present an LP-rounding 2-approximation algorithm for a flexible version of the Generalized Assignment Problem that also applies to 1-FSRM. Finally, we give an \(\varOmega (\frac{r}{\log r})\) lower bound on the approximation ratio for r-FSRM (assuming P \(\ne \) NP).

Flexible allocation on related machines with assignment restrictions

Abstract We introduce the problem of Flexible Allocation on Related Machines with Assignment Restrictions ( FARM ). In this problem the input consists of a set of machines and a set of jobs. Each machine has a finite capacity, and each job has a resource requirement interval, a profit per allocated unit of resource, and a set of machines that can potentially supply the requirement. A feasible solution is an allocation of machine resources to jobs such that: (i) a machine resource can be allocated to a job only if it is a potential supplier of this job, (ii) the amount of machine resources allocated by a machine is bounded by its capacity, and (iii) the amount of resources that are allocated to a job is either in its requirement interval or zero. Notice that a job can be serviced by multiple machines. The goal is to find a feasible allocation that maximizes the overall profit. We focus on r - FARM in which the required resource of a job is at most an r -fraction of (or r times) the capacity of each potential machine, for some r ≥ 0 . FARM is motivated by resource allocation problems arising in cellular networks and in cloud computing. Specifically, FARM models the problem of assigning clients to base stations in 4G cellular networks. We present a 2-approximation algorithm for 1- FARM and a 1 1 − r -approximation algorithm for r - FARM , for any r ∈ ( 0 , 1 ) . Both are based on the local ratio technique and on maximum flow computations. We observe that an LP-rounding 2-approximation algorithm for the Generalized Assignment Problem also applies to 1- FARM . Finally, we give an Ω ( r log r ) lower bound on the approximation ratio for r - FARM (assuming P ≠ NP).

Optimizing bandwidth allocation in elastic optical networks with application to scheduling

Abstract We study a problem of optimal bandwidth allocation in the elastic optical networks technology, where usable frequency intervals are of variable width. In this setting, each lightpath has a lower and upper bound on the width of its frequency interval, as well as an associated profit, and we seek a bandwidth assignment that maximizes the total profit. This problem is known to be NP-complete. We strengthen this result by showing that, in fact, the problem is inapproximable within any constant ratio even on a path network. We further derive NP-hardness results and present approximation algorithms for several special cases of the path and ring networks, which are of practical interest. Finally, while in general our problem is hard to approximate, we show that an optimal solution can be obtained by allowing resource augmentation . Some of our results resolve open problems posed by Shalom et al. (2013) [28] . Our study has applications also in real-time scheduling.