This paper examines two di erent mechanisms for saving power in battery-operated embedded systems. The rst is that the system can be placed in a sleep state if it is idle. However, a xed amount of energy is required to bring the system back i n to an active state in which it can resume work. The second way i n w h i c h p o wer savings can be achieved is by v arying the speed at which jobs are run. We utilize a power consumption curve P (s) w h i c h indicates the power consumption level given a particular speed. We assume that P (s) i s c o n vex, non-decreasing and non-negative f o r s 0. The problem is to schedule arriving jobs in a way that minimizes total energy use and so that each job is completed after its release time and before its deadline. We assume that all jobs can be preempted and resumed at no cost. Although each problem has been considered separately, this is the rst theoretical analysis of systems that can use both mechanisms. We g i v e an o ine algorithm that is within a factor of two of the optimal algorithm. We a l s o g i v e an online algorithm with a constant competitive ratio.

We study the complexity of a class of problems involving satisfying constraints which remain the same under translations in one or more spatial directions. In this paper, we show hardness of a classical tiling problem on an N ×N 2-dimensional grid and a quantum problem involving finding the ground state energy of a 1-dimensional quantum system of N particles. In both cases, the only input is N , provided in binary. We show that the classical problem is NEXP-complete and the quantum problem is QMA EXPcomplete. Thus, an algorithm for these problems which runs in time polynomial in N (exponential in the input size) would imply that EXP = NEXP or BQEXP = QMA EXP , respectively. Although tiling in general is already known to be NEXP-complete, to our knowledge, all previous reductions require that either the set of tiles and their constraints or some varying boundary conditions be given as part of the input. In the problem considered here, these are fixed, constant-sized parameters of the problem. Instead, the problem instance is encoded solely in the size of the system.

Online dynamic power management (DPM) strategies refer to strategies that attempt to make power-mode-related decisions based on information available at runtime. In making such decisions, these strategies do not depend upon information of future behavior of the system, or any a priori knowledge of the input characteristics. In this paper, we present online strategies, and evaluate them based on a measure called the competitive ratio that enables a quantitative analysis of the performance of online strategies. All earlier approaches (online or predictive) have been limited to systems with two power-saving states (e.g., idle and shutdown). The only earlier approaches that handled multiple power-saving states were based on stochastic optimization. This paper provides a theoretical basis for the analysis of DPM strategies for systems with multiple power-down states, without resorting to such complex approaches. We show how a relatively simple "online learning" scheme can be used to improve the competitive ratio over deterministic strategies using the notion of "probability-based" online DPM strategies. Experimental results show that the algorithm presented here attains the best competitive ratio in comparison with other known predictive DPM algorithms. The other algorithms that come close to matching its performance in power suffer at least an additional 40% wake-up latency on average. Meanwhile, the algorithms that have comparable latency to our methods use at least 25% more power on average.

We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a one-dimensional quantum system (with 9 states per particle). This might have practical implications for experimentalists interested in constructing an adiabatic quantum computer. Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum particles is QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to the fact that the analogous classical problem, namely, one dimensional MAX-2-SAT with nearest neighbor constraints, is in P. The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Not all illegal configurations can be ruled out by local checks, so instead we rule out such illegal configurations because they would, in the future, evolve into a state which can be seen locally to be illegal. Since it is unlikely that quantum computers can efficiently solve QMA problems, our construction gives a one-dimensional system which takes an exponential time to relax to its ground state at any temperature. This makes it a candidate for a one-dimensional spin glass.