We consider Lagrangian duality based approaches to design and analyze algorithms for online energy-efficient scheduling. First, we present a primal-dual framework. Our approach makes use of the Lagrangian weak duality and convexity to derive dual programs for problems which could be formulated as convex assignment problems. The duals have intuitive structures as the ones in linear programming. The constraints of the duals explicitly indicate the online decisions and naturally lead to competitive algorithms. Second, we use a dual-fitting approach, which also based on the weak duality, to study problems which are unlikely to admit convex relaxations. Through the analysis, we show an interesting feature in which primal-dual gives idea for designing algorithms while the analysis is done by dual-fitting. We illustrate the advantages and the flexibility of the approaches through problems in different setting: from single machine to unrelated machine environments, from typical competitive analysis to the one with resource augmentation, from convex relaxations to non-convex relaxations.
In the search for principled methods to design and analyze online problems, especially in online scheduling, interesting approaches @cite_1 @cite_11 @cite_6 based on mathematical programming have been presented. The approaches give insight about the nature of many scheduling problems, hence lead to algorithms which are usually simple and competitive @cite_1 @cite_11 @cite_6 @cite_17 @cite_8 @cite_5 .
@cite_11 @cite_8 @cite_1 @cite_6 @cite_5 @cite_17
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We give a principled method to design online algorithms (for potentially non-linear problems) using a mathematical programming formulation of the problem, and also to analyze the competitiveness of the resulting algorithm using the dual program. This method can be viewed as an extension of the online primal-dual method for linear programming problems, to nonlinear programs. We show the application of this method to two online speed-scaling problems: one involving scheduling jobs on a speed scalable processor so as to minimize energy plus an arbitrary sum scheduling objective, and one involving routing virtual circuit connection requests in a network of speed scalable routers so as to minimize the aggregate power or energy used by the routers. This analysis shows that competitive algorithms exist for problems that had resisted analysis using the dominant potential function approach in the speed-scaling literature, and provides alternate cleaner analysis for other known results. This gives us another tool in the design and analysis of primal-dual algorithms for online problems. The main aim of this note is to show that a polynomial-time algorithm for the scheduling problem 1|r j ; p j = p| ∑ Uj given by Carlier in (1981) is incorrect. In this problem we are given n jobs with release times and deadlines. All jobs have the same processing time p. The objective is to find a non-preemptive schedule that maximizes the number of jobs completed by their deadlines. The feasibility version of this problem, where we ask whether all jobs can meet their deadlines, has been studied thoroughly. Polynomial-time algorithms for this version were first found, independently, by Simons (1978) and Carlier (1981). A faster algorithm, with running time O(n log n), was subsequently given by (1981). The elegant feasibility algorithm of Carlier (1981) is based on dynamic programming and it processes jobs from left to right on the time-axis. For each time t, it constructs a partial schedule with jobs that complete at or before time t. Carlier also attempted to apply the same technique to design a polynomial-time algorithm for the maximization version, 1|r j ; p j = p| ∑ Uj , and claimed an O(n3 log n)-time algorithm. His result is now widely cited in the literature. We show, however, that this algorithm is not correct, by giving an instance on which it produces a sub-optimal schedule. Our counter-example can be, in fact, extended to support a broader claim, namely that even the general approach from Carlier (1981) cannot yield a polynomial-time algorithm. By this general approach we mean a class of algorithms that processes the input from left to right and make decisions based on the deadline ordering, and not their exact values. The question remains as to how efficiently can we solve the scheduling problem 1|r j ; p j = p| ∑ Uj . Baptiste (1999) gave an O(n7)-time algorithm for the more general version of this problem where jobs have weights. We show how to modify his algorithm to obtain a faster, O(n5)-time algorithm for the non-weighted case. These last two results are discussed only briefly in this note. The complete proofs can be found in the full version of this paper, see (2004). We present an unified approach to study online scheduling problems in the resource augmentation speed scaling models. Potential function method is extensively used for analyzing algorithms in these models; however, they yields little insight on how to construct potential functions and how to design algorithms for related problems. In the paper, we generalize and strengthen the dual-fitting technique proposed by [1]. The approach consists of considering a possibly non-convex relaxation and its Lagrangian dual; then constructing dual variables such that the Lagrangian dual has objective value within a desired factor of the primal optimum. The competitive ratio follows by the standard Lagrangian weak duality. This approach is simple yet powerful and it is seemingly a right tool to study problems with resource augmentation or speed scaling. We illustrate the approach through the following results. 1 We revisit algorithms EQUI and LAPS in Non-clairvoyant Scheduling to minimize total flow-time. We give simple analyses to prove known facts on the competitiveness of such algorithms. Not only are the analyses much simpler than the previous ones, they also explain why LAPS is a natural extension of EQUI to design a scalable algorithm for the problem. 2 We consider the online scheduling problem to minimize total weighted flow-time plus energy where the energy power f(s) is a function of speed s and is given by s α for α ≥ 1. For a single machine, we showed an improved competitive ratio for a non-clairvoyant memoryless algorithm. For unrelated machines, we give an O(α logα)-competitive algorithm. The currently best algorithm for unrelated machines is O(α 2)-competitive. 3 We consider the online scheduling problem on unrelated machines with the objective of minimizing ∑ i,j w ij f(F j ) where F j is the flow-time of job j and f is an arbitrary non-decreasing cost function with some nice properties. We present an algorithm which is ( 1 1-3 )-speed, ( 2K( ) )-competitive where K(e) is a function depending on f and e. The algorithm does not need to know the speed (1 + e) a priori. A corollary is a (1 + e)-speed, ( k ^ 1+1 k )-competitive algorithm (which does not know e a priori) for the objective of minimizing the weighted l k -norm of flow-time. The problem considered here is the same as the one discussed in [G. Galambos and G. J. Woeginger, eds., SIAM J. Comput., 22 (1993), pp. 349--355]. It is an m-machine online scheduling problem in which we wish to minimize the competitive ratio for the makespan objective. In this paper, we show that @math is a lower bound on this competitive ratio for m=4. In particular, we show how to force a lower bound of @math for any positive @math . This reduces the gap between the performance of known algorithms [S. Albers, in Proceedings of the 29th Annual ACM Symposium on Theory of Computing, ACM, New York, 1997, pp. 130--139] and the lower bound. The method used introduces an approach to building the task master's strategy.