Mordechai Shalom

A 10/7 + /spl epsi/ approximation for minimizing the number of ADMs in SONET rings

SONET ADMs are dominant cost factors in WDM/SONET rings. Whereas most previous papers on the topic concentrated on the number of wavelengths assigned to a given set of lightpaths, more recent papers argue that the number of ADMs is a more realistic cost measure. Some of these works discuss various heuristic algorithms for this problem, and the best known result is a 3/2 approximation in G. Calinescu and P.J. Wan (2001). Through the study of the relation between this problem and the problem of finding maximum disjoint rings in a given set of lightpaths we manage to shed more light onto this problem and to develop a 10/7 + /spl epsi/ approximation for it.

Minimizing total busy time in parallel scheduling with application to optical networks

We consider a scheduling problem in which a bounded number of jobs can be processed simultaneously by a single machine. The input is a set of n jobs J = {J1, … , Jn}. Each job, Jj, is associated with an interval [sj, cj] along which it should be processed. Also given is the parallelism parameter g ≥ 1, which is the maximal number of jobs that can be processed simultaneously by a single machine. Each machine operates along a contiguous time interval, called its busy interval, which contains all the intervals corresponding to the jobs it processes. The goal is to assign the jobs to machines such that the total busy time of the machines is minimized. The problem is known to be NP-hard already for g = 2. We present a 4-approximation algorithm for general instances, and approximation algorithms with improved ratios for instances with bounded lengths, for instances where any two intervals intersect, and for instances where no interval is properly contained in another. Our study has important application in optimizing the switching costs of optical networks.

Approximating the Traffic Grooming Problem with Respect to ADMs and OADMs

We consider the problem of switching cost in optical networks, where messages are sent along lightpaths. Given lightpaths, we have to assign them colors, so that at most glightpaths of the same color can share any edge (gis the grooming factor). The switching of the lightpaths is performed by electronic ADMs (Add-Drop-Multiplexers) at their endpoints and optical ADMs (OADMs) at their intermediate nodes. The saving in the switching components becomes possible when lightpaths of the same color can use the same switches. Whereas previous studies concentrated on the number of ADMs, we consider the cost function - incurred also by the number of OADMs - of f(i¾?) = i¾?|OADMs| + (1 i¾? i¾?)|ADMs|, where 0 ≤ i¾?≤ 1. We concentrate on chain networks, but our technique can be directly extended to ring networks. We show that finding a coloring which will minimize this cost function is NP-complete, even when the network is a chain and the grooming factor is g= 2, for any value of i¾?. We then present a general technique that, given an r-approximation algorithm working on particular instances of our problem, i.e. instances in which all requests share a common edge of the chain, builds a new algorithm for general instances having approximation ratio ri¾?logni¾?. This technique is used in order to obtain two polynomial time approximation algorithms for our problem: the first one minimizes the number of OADMs (the case of i¾?= 1), and its approximation ratio is 2 i¾?logni¾?; the second one minimizes the combined cost f(i¾?) for 0 ≤ i¾?< 1, and its approximation ratio is

Optimizing busy time on parallel machines

2 \sqrt{g}~\lceil \log n \rceil

Optimizing Busy Time on Parallel Machines

.

On minimizing the number of ADMs in a general topology optical network

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.

Online optimization of 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.

On the complexity of the traffic grooming problem in optical networks

Minimizing the number of electronic switches in optical networks is a main research topic in recent studies. In such networks we assign colors to a given set of lightpaths. Thus the lightpaths are partitioned into cycles and paths, and the switching cost is minimized when the number of paths is minimized. The problem of minimizing the switching cost is NP-hard. A basic approximation algorithm for this problem eliminates cycles of size at most l, and has a performance guarantee of

Profit Maximization in Flex-Grid All-Optical Networks

OPT+\frac{1}{2}(1+\epsilon)N

Approximating the traffic grooming problem

, where OPT is the cost of an optimal solution, N is the number of lightpaths, and