On the competitive ratio of evaluating priced functions

We study competitive function evaluation in the context of computing with priced information. A function f has to be evaluated for a fixed but unknown choice of the values of the variables. Each variable x of f has an associated cost c(x), which has to be paid to read the value of x. The problem is to design algorithms that compute the function querying the values of the variables sequentially while trying to minimize the total cost incurred. The evaluation of the performance of the algorithms is made by employing competitive analysis. We determine the best possible extremal competitive ratio for the classes of threshold trees, game trees, and monotone boolean functions with constrained minterms, by providing a polynomial-time algorithm whose competitiveness matches the known lower bounds.

We consider the problem of learning monotone Boolean functions over {0, 1}/sup n/ under the uniform distribution. Specifically, given a polynomial number of uniform random samples for an unknown monotone Boolean function f, and given polynomial completing time, we would like to approximate f as well as possible. We describe a simple algorithm that we prove achieves error at most 1/2-/spl Omega/(1//spl radic/n), improving on the previous best bound of 1/2-/spl Omega/((log/sup 2/ n)/n). We also prove that no algorithm, given a polynomial number of samples, can guarantee error 1/2-/spl omega/((log n)//spl radic/n), improving on the previous best hardness bound of O(1//spl radic/n). These lower bounds hold even if the learning algorithm is allowed membership queries. Thus this paper settles to an O(log n) factor the question of the best achievable error for learning the class of monotone Boolean functions with respect to the uniform distribution.

The paper “An Optimal Control Framework for Online Job Scheduling with General Cost Functions,” by Etesami devises online speed-augmented competitive algorithms for minimizing the generalized completion time on a single and multiple unrelated machines for a very general class of cost functions. To that end, a novel optimal control formulation for the offline version of the problem is developed. Such a formulation allows using tools from optimal control theory, such as the minimum principle and Hamilton-Jacobi-Bellman equation, to set the dual variables as close as possible to the optimal dual variables and leverage them to design primal-dual online algorithms as an iterative application of the offline problem. The analysis can achieve state-of-the-art competitive ratios for several special cases and provide new competitive ratios which are the first in their settings. In particular, the analysis offers a principled method of estimating dual variables in a general setting of online job scheduling.

4 - Recursive Enumerability

Weighted group search on a line & implications to the priority evacuation problem

We present two deterministic online algorithms for the problem of scheduling with a general machine cost function. In this problem, every machine has a cost that is given by a nonnegative cost function, and the objective function is the sum of makespan and the cost of the purchased machines. In previous work by Imreh, it was shown that no deterministic algorithm has competitive ratio below 2, and an algorithm whose competitive ratio does not exceed $$(3+\sqrt{5})/2 \approx 2.618$$ was presented. Our first algorithm imitates an optimal offline solution with respect to the number of machines used, and we show that its competitive ratio is exactly 2.5. Then, we modify our algorithm by imitating a preemptive optimal offline solution instead of a non-preemptive one. This leads to the design of a 2-competitive algorithm, which is the best possible.

At present, problems related to data mining are aris-ing in various poorly formalizable areas of research(genetics, archaeology, medicine, etc.). Such problemsare characterized by insufficient knowledge about theobjects under study, which hampers the development ofmathematical models for them; by a large number ofmultitype (quantitative and qualitative) factors and arelatively small amount of data; and by the requirementthat analysis results be represented in a form clear toexperts in applied areas.A promising approach to solving such problems isbased on the class of logical decision functions. Theyare frequently represented in the form of a decisiontree. A detailed description of this class and referencescan be found in [1].The construction of decision functions with a mini-mal risk of erroneous predictions is an important task inmethods based on logical decision functions and inother data analysis techniques. It is well known that thecomplexity of the class of decision functions (thecapacity characteristic of the class [2]) is a importantfactor affecting the quality of decisions. To achieve thebest quality, we need a certain tradeoff between thecomplexity of the class and the accuracy of decisionsobtained on the training sample. Thus, the problemarises of choosing an optimal complexity of the class.In numerous applications, empirical data (trainingsample) are supplemented with expert knowledge,which is not tightly related to a distribution model(expert knowledge can be specified as upper bounds forthe risk; they can express constraints on the class of dis-tributions or the class of decision functions, specifypreference rules used in decision making, etc.). Whenan optimal complexity of the class is chosen, availableempirical data and expert knowledge have to be jointlytaken into account.This problem can be solved within the framework ofthe Bayesian learning theory. The idea behind thisapproach is to use a priori knowledge about the prob-lem in question so that each possible distribution (strat-egy, the state of nature) is assigned a certain weight.This weight reflects the expert’s intuitive confidence inwhether the unknown true distribution coincides withthat under consideration.In this paper, we propose a Bayesian recognitionmodel based on a finite set of events. The model is usedto develop and analyze methods for constructing logi-cal decision functions. While choosing an optimal com-plexity of the class, the model takes into account empir-ical data and expert knowledge.In pattern recognition, the task is to predict the pat-tern’s index for an arbitrary object of the populationdescribed by a set of variables. The prediction is madeby analyzing a training sample consisting of the valuesof these variables, together with the index of the corre-sponding pattern for each object. The variables can beof different types; i.e., some of them can be quantita-tive, while the others can be qualitative. As a rule, theproblem is solved in a class of decision functions inwhich an optimal function is sought according to agiven criterion. The class of logical decision functionsis defined on a set of partitions of the feature space intoa finite number of subdomains described by conjunc-tions of simple predicates. The number of subdomainsdetermines the degree of complexity of a logical func-tion.A Bayesian pattern recognition model on a finite setof events is defined by formulating certain statementsthat avoid the local metric properties of the featurespace (the transition from points of the space to events,where an event is understood as that of the original vari-ables taking values from a certain subdomain of thepartition). Moreover, the pattern recognition problem isconsidered on the values of a discrete unordered vari-able, the concept of a learning method (a mapping from

Three well studied progress conditions for implementing concurrent algorithms without locking are, obstruction-freedom, non-blocking and wait-freedom. Obstruction-freedom is weaker than non-blocking which, in turn, is weaker than wait-freedom. While obstruction-freedom and non-blocking have the potential to significantly improve the performance of concurrent applications, wait-freedom (although desirable) imposes too much overhead upon the implementation. In [5], Fich, Luchangco, Moir, and Shavit have presented an interesting transformation that converts any obstruction-free algorithm into a wait-free algorithm when analyzed in the unknown-bound semi-synchronous model. The FLMS transformation uses n atomic single-writer registers, n atomic multi-writer registers and a single fetch-and-increment object, where n is the number of processes. We define a time complexity measure for analyzing such transformations, and prove that the time complexity of the FLMS transformation is exponential in the number of processes n. This leads naturally to the question of whether the time and/or space complexity of the FLMS transformation can be improved by relaxing the wait-freedom progress condition. We present several efficient transformations that convert any obstruction-free algorithm into a non-blocking algorithm when analyzed in the unknown-bound semi-synchronous model. All our transformations have O(1) time complexity. One transformation uses n atomic single-writer registers and a single compare-and-swap object; another transformation uses only a single compare-and-swap object which is assumed to support also a read operation.

A special class of monotone Boolean functions coming from shadow minimization theory of finite set-systems – KK-MBF functions – is considered. These functions are a descriptive model for systems of compatible groups of constraints, however, the class of all functions is unambiguously complex and it is sensible to study relatively simple subclasses of functions such as KK-MBF. Zeros of KK-MBF functions correspond to initial segments of lexicographic order on hypercube layers. This property is used to simplify the recognition. Lexicographic order applies priorities over constraints which is applicable property of practices. Query-based algorithms for KK-MBF functions are investigated in terms of their complexities.

Full-chip statistical leakage power analysis typically requires quadratic time complexity in the presence of spatial correlation. When spatial correlation are strong (with large spatial correlation length), efficient linear time complexity analysis can be attained as the number of variational variables can be significantly reduced. However this is not the case for circuits where gate leakage currents are weakly correlated. In this paper, we present a linear time algorithm for statistical leakage power analysis in the presence of weak spatial correlation. The new algorithm exploits the fact that gate leakage current can be efficiently computed locally when correlation is weak. We adopt a newly proposed spatial correlation model where a new set of location-dependent uncorrelated variables are defined over virtual grids to represent the original physical random variables via fitting. To compute the leakage current of a gate on the new set of variables, the new method uses the orthogonal polynomials based collocation method, which can be applied to any gate leakage models. The total leakage currents are then computed by simply summing up the resulting orthogonal polynomials (their coefficients) on the new set of variables for all gates. Experimental results show that the proposed method is about two orders of magnitude faster than the recently proposed grid-based method [3] with similar accuracy and many orders of magnitude times over the Monte Carlo method.

Recently, increasing penetration of renewable energy generation has created challenges for power system operators to perform efficient power generation daily scheduling, due to the intermittent nature of the renewable generation and discrete decisions of each generation unit. Among all aspects to be considered, a unit commitment polytope is fundamental and embedded in the models at different stages of power system planning and operations. In this article, we focus on deriving polynomial-time algorithms for the unit commitment problems with a general convex cost function and piecewise linear cost function, respectively. We refine an time, where T represents the number of time periods, algorithm for the deterministic single-generator unit commitment problem with a general convex cost function and accordingly develop an extended formulation in a higher-dimensional space that can provide an integral solution, in which the physical meanings of the decision variables are described. This means the original problem can be solved as a convex program instead of a mixed-integer convex program. Furthermore, for the case in which the cost function is piecewise linear, by exploring the optimality conditions, we derive more efficient algorithms for both deterministic (i.e., time) and stochastic (i.e., time, where N represents the number of nodes in the stochastic scenario tree) single-generator unit commitment problems. We also develop the corresponding extended formulations for both deterministic and stochastic single-generator unit commitment problems that solve the original mixed-integer linear programs as linear programs. Similarly, physical meanings of the decision variables are explored to show the insights of the new modeling approach.

A typical multivariable control problem requires the design of dynamic compensation to guarantee the following desirable behavior of the closed loop system. 1. Each of an assigned set of output variables converges to, or tracks, a corresponding observed reference input from a specified function class. This is the servo problem. 2. Each of an assigned set of output variables converges to zero from arbi-trary initial values, when the system is perturbed by possibly nonmeasur-able disturbances in a specified function class. This is the regulator problem; the convergent variables are said to be regulated; and zero represents the desired or set-point value.

In this thesis, we study the theoretical approach on energy-efficient scheduling problems arising in demand response management in the modern electrical smart grid. Consumers send in power requests with flexible feasible timeslots during which their requests can be served. The grid controller, upon receiving power requests, schedules each request within the specified interval. The electricity cost is measured by a convex function of the load in each timeslot. The objective is to schedule all requests with the minimum total electricity cost. We study the smart grid scheduling problem in different models. For the offline model, we prove the problem is NP-hard for the general case. We propose a polynomial time algorithm for special input where jobs have unit power request and unit time duration. By adapting the polynomial time algorithm for unit-size jobs, we propose an approximation algorithm for more general input. On the other hand, we also present an exact algorithm to find the optimal schedule for the problem with general input. For the online model, we propose an online algorithm for jobs with jobs with arbitrary power request, arbitrary time duration, and arbitrary contiguous feasible intervals. We also show a lower bound of the competitive ratio for the smart grid scheduling problem with unit height and arbitrary width. For special cases, we design different online algorithms with better competitive ratios. Finally, we look at other optimization problems and show how to solve them by adapting our techniques. We prove that our online algorithm can solve the machine minimization problem with an asymptotically optimal competitive ratio. We also show that our exact algorithm can be adapted to solve other demand response management problems.

The neural control of movement has been described using different sets of elemental variables. Two possible sets of elemental variables have been suggested for finger pressing tasks: the forces of individual fingers and the finger commands (also called finger modes or central commands). The authors analyzed which of the 2 sets of the elemental variables is more likely used in the optimization of the finger force sharing and which set is used for the stabilization of performance. They used two recently developed techniques—the analytical inverse optimization (ANIO) and the uncontrolled manifold (UCM) analysis—to evaluate each set of elemental variables with respect to both aspects of performance. The results of the UCM analysis favored the finger commands as the elemental variables used for performance stabilization, while ANIO worked equally well on both sets of elemental variables. A simple scheme is suggested as to how the CNS could optimize a cost function dependent on the finger forces, but for the sake of facilitation of the feed forward control it substitutes the original cost function by a cost function, which is convenient to optimize in the space of finger commands.

Regular cost functions have been introduced recently as an extension to the notion of regular languages with counting capabilities. The specificity of cost functions is that exact values are not considered, but only estimated. In this paper, we study the strict subclass of regular temporal cost functions. In such cost functions, it is only allowed to count the number of occurrences of consecutive events. For this reason, this model intends to measure the length of intervals, i.e., a discrete notion of time. We provide various equivalent representations for functions in this class, using automata, and 'clock based' reduction to regular languages. We show that the conversions are much simpler to obtain, and much more efficient than in the general case of regular cost functions. Our second aim in this paper is to use temporal cost function as a test-case for exploring the algebraic nature of regular cost functions. Following the seminal ideas of Schutzenberger, this results in a decidable algebraic characterization of regular temporal cost functions inside the class of regular cost functions.