We consider the makespan minimization coupled-tasks problem in presence of
compatibility constraints with a specified topology. In particular, we focus on
stretched coupled-tasks, i.e. coupled-tasks having the same sub-tasks execution
time and idle time duration. We study several problems in framework of classic
complexity and approximation for which the compatibility graph is bipartite
(star, chain,. . .). In such a context, we design some efficient
polynomial-time approximation algorithms for an intractable scheduling problem
according to some parameters
We present a PTAS for the Multiple Subset Sum Problem (MSSP) with different knapsack capacities. This is the selection of items from a given ground set and their assignment to a given number of knapsacks such that the sum of the item weights in every knapsack does not exceed its capacity and the total sum of the weights of the packed items is as large as possible. Our result generalizes the PTAS for the special case in which all knapsack capacities are identical [1]
The rise of next-generation sequencing has produced an abundance of data with almost limitless analysis applications. As sequencing technology decreases in cost and increases in throughput, the amount of available data is quickly outpacing improve- ments in processor speed. Analysis methods must also increase in scale to remain computationally tractable. At the same time, larger datasets and the availability of population-wide data offer a broader context with which to improve accuracy.
This thesis presents three tools that improve the scalability of sequencing data storage and analysis. First, a lossy compression method for RNA-seq alignments offers extreme size reduction without compromising downstream accuracy of isoform assembly and quantitation. Second, I describe a graph genome analysis tool that filters population variants for optimal aligner performance. Finally, I offer several methods for improving CNV segmentation accuracy, including borrowing strength across samples to overcome the limitations of low coverage. These methods compose a practical toolkit for improving the computational power of genomic analysis
In this thesis we approach several problems with approximation algorithms; these are feasibility problems as well as optimization problems. In Chapter 1 we give a brief introduction into the general paradigm of approximation algorithms, motivate the problems, and give an outline of the
thesis. In Chapter 2, we discuss two algorithms to approximately generate a feasible solution of the mixed packing and covering problem which is
a model from convex optimization. This problem includes a large class of
linear programs. The algorithms generate approximately feasible solutions
within O(M(ln M+epsilon^{-2} ln epsilon^{-1})) and
O(M epsilon{-2} ln (M epsilon^{-1}))iterations,respectively,whereineachiterationablockproblemwhichdependsonthespecificapplicationhastobesolved.Bothalgorithms,appliedtolinearprograms,canresultincolumngenerationalgorithms.InChapter3,weimplementanalgorithmfortheso−calledmax−min−resourcesharingproblem.Thisisacertainconvexoptimizationproblemwhich,similartotheprobleminChapter1,includesalargeclassoflinearprograms.Theimplementation,whichisincludedintheappendix,isdoneinC++.WeusetheimplementationinthecontextofanAFPTASforStripPackinginordertoevaluatedynamicoptimizationofaparameterinthealgorithm,namelythesteplengthusedforinterpolation.Wecompareourchoicetothestaticsteplengthproposedintheanalysisofthealgorithmandconcludethatdynamicoptimizationofthesteplengthsignificantlyreducesthenumberofiterations.InChapter4,westudytwocloselyrelatedschedulingproblems,namelynon−preemptiveschedulingwithfixedjobsandschedulingwithnon−availabilityforsequentialjobsonmidenticalmachinesunderthemakespanobjective,wheremisconstant.Forthefirstproblem,whichdoesnotadmitanFPTASunlessP=NP,weobtainanewPTAS.Forthesecondproblem,weshowthatasuitablerestriction(namelythepermanentavailabilityofonemachine)isnecessarytoobtainaboundedapproximationratio.Forthisrestriction,whichdoesnotadmitanFPTASunlessP=NP,wepresentaPTAS;wealsodiscussthecomplexityofvariousspecialcases.Intotal,theresultsarebasicallybestpossible.InChapter5,wecontinuethestudiesfromChapter4wherenowthenumbermofmachinesispartoftheinput,whichmakestheproblemalgorithmicallyharder.Schedulingwithfixedjobsdoesnotadmitanapproximationratiobetterthan3/2,unlessP=NP;hereweobtainanapproximationratioof3/2+epsilonforanyepsilon>0.Forschedulingwithnon−availability,werequireaconstantpercentageofthemachinestobepermanentlyavailable.Thisrestrictionalsodoesnotadmitanapproximationratiobetterthan3/2unlessP=NP;wealsoobtainanapproximationratioof3/2+\epsilon$ for any epsilon>0. With an interesting argument, the approximation ratio for both problems is
refined to exactly 3/2. We also point out an interesting relation of scheduling with fixed jobs to Bin Packing. As in Chapter 4, the results are in a certain sense best possible.
Finally, in Chapter 6, we conclude
with some remarks and open research problems