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Abstract. Given a complex matrix H, we consider the decomposition H = QRP * , where R is upper triangular and Q and P have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where R is an upper triangular matrix with the eigenvalues of H on the diagonal. We show that any diagonal for R can be achieved that satisfies Weyl's multiplicative majorization conditions:where K is the rank of H, σ i is the i-th largest singular value of H, and r i is the i-th largest (in magnitude) diagonal element of R. Given a vector r which satisfies Weyl's conditions, we call the decomposition H = QRP * , where R is upper triangular with prescribed diagonal r, the generalized triangular decomposition (GTD). A direct (nonrecursive) algorithm is developed for computing the GTD. This algorithm starts with the SVD and applies a series of permutations and Givens rotations to obtain the GTD. The numerical stability of the GTD update step is established. The GTD can be used to optimize the power utilization of a communication channel, while taking into account quality of service requirements for subchannels. Another application of the GTD is to inverse eigenvalue problems where the goal is to construct matrices with prescribed eigenvalues and singular values.

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Motivated by issues of saving energy in data centers we define a collection of new problems referred to as "machine activation" problems. The central framework we introduce considers a collection of m machines (unrelated or related) with each machine i having an activation cost of ai. There is also a collection of n jobs that need to be performed, and pi,j is the processing time of job j on machine i. Standard scheduling models assume that the set of machines is fixed and all machines are available. However, in our setting, we assume that there is an activation cost budget of A -we would like to select a subset S of the machines to activate with total cost a(S) ≤ A and find a schedule for the n jobs on the machines in S minimizing the makespan (or any other metric).We consider both the unrelated machines setting, as well as the setting of scheduling uniformly related parallel machines, where machine i has activation cost ai and speed si, and the processing time of job j on machine i is pi,j = p j s i , where pj is the processing requirement of job j.For the general unrelated machine activation problem, our main results are that if there is a schedule with makespan T and activation cost A then we can obtain a schedule with makespan (2+ )T and activation cost 2(1+ 1 )(ln n OP T +1)A, for any > 0. We also consider assignment costs for jobs as in the generalized assignment problem, and using our framework, provide algorithms that minimize the machine activation and the assignment cost simultaneously. In addition, we present a greedy algorithm which only works for the basic version and yields a makespan of 2T and an activation cost A(1 + ln n).For the uniformly related parallel machine scheduling problem, we develop a polynomial time approximation scheme that outputs a schedule with the property that the activation cost of the subset of machines is at most A and the makespan is at most (1 + )T for any > 0. For the special case of m identical speed machines, the machine activation problem is trivial, since the cheapest subset of k machines is always the best choice if the optimal solution activates k machines. In addition, we consider the case when some jobs can be dropped (and are treated as outliers).