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.
presented the primal-dual method for online packing and covering problems. Their method unifies several previous potential function based analyses and is a powerful tool to design and analyze algorithms for problems with linear relaxations. gave a primal-dual algorithm for a general class of scheduling problems with cost function @math . also used the primal-dual approach to derive optimal competitive ratios for online matching with concave return. The construction of dual programs in @cite_17 @cite_9 is based on convex conjugates and Fenchel duality for primal convex programs in which the objective is convex and the constraints are .
@cite_9 @cite_17
1988278843 2272870368
We consider a significant generalization of the Adwords problem by allowing arbitrary concave returns, and we characterize the optimal competitive ratio achievable. The problem considers a sequence of items arriving online that have to be allocated to agents, with different agents bidding different amounts. The objective function is the sum, over each agent i, of a monotonically non-decreasing concave function Mi : R+ -> R+ of the total amount allocated to i. All variants of online matching problems (including the Adwords problem) studied in the literature consider the special case of budgeted linear functions, that is, functions of the form Mi(ui) = min ui,Bi for some constant Bi. The distinguishing feature of this paper is in allowing arbitrary concave returns. The main result of this paper is that for each concave function M, there exists a constant F(M) ≤ 1 such that: there exists an algorithm with competitive ratio of miniF(Mi), independent of the sequence of items. No algorithm has a competitive ratio larger than F(M) over all instances with Mi= M for all i. Our algorithm is based on the primal-dual paradigm and makes use of convex programming duality. The upper bounds are obtained by formulating the task of finding the right counterexample as an optimization problem. This path takes us through the calculus of variations which deals with optimizing over continuous functions. The algorithm and the upper bound are related to each other via a set of differential equations, which points to a certain kind of duality between them. We consider the problem of online scheduling of jobs on unrelated machines with dynamic speed scaling to minimize the sum of energy and weighted flow time. We give an algorithm with an almost optimal competitive ratio for arbitrary power functions. (No earlier results handled arbitrary power functions for minimizing flow time plus energy with unrelated machines.) For power functions of the form f(s) = sα for some constant α > 1, we get a competitive ratio of O(α log α), improving upon a previous competitive ratio of O(α2) by [3], along with a matching lower bound of Ω(α log α). Further, in the resource augmentation model, with a 1 + e speed up, we give a 2(1 e + 1) competitive algorithm, with essentially the same techniques, improving the bound of 1 + O(1 e2) by [15] and matching the bound of [3] for the special case of fixed speed unrelated machines. Unlike the previous results most of which used an amortized local competitiveness argument or dual fitting methods, we use a primal-dual method, which is useful not only to analyze the algorithms but also to design the algorithm itself.