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A New Approach to Online Scheduling: Approximating the Optimal Competitive Ratio
"... We propose a new approach to competitive analysis in online scheduling by introducing the novel concept of competitiveratio approximation schemes. Such a scheme algorithmically constructs an online algorithm with a competitive ratio arbitrarily close to the best possible competitive ratio for any o ..."
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We propose a new approach to competitive analysis in online scheduling by introducing the novel concept of competitiveratio approximation schemes. Such a scheme algorithmically constructs an online algorithm with a competitive ratio arbitrarily close to the best possible competitive ratio for any online algorithm. We study the problem of scheduling jobs online to minimize the weighted sum of completion times on parallel, related, and unrelated machines, and we derive both deterministic and randomized algorithms which are almost best possible among all online algorithms of the respective settings. We also generalize our techniques to arbitrary monomial cost functions and apply them to the characterization of online algorithms combined with various simplifications and transformations. We also contribute algorithmic means to compute the actual value of the best possible competitive ratio up to an arbitrary accuracy. This strongly contrasts (nearly) all previous manually obtained competitiveness results and, most importantly, it reduces the search for the optimal competitive ratio to a question that a computer can answer. We believe that our concept can also be applied to many other problems and yields a new perspective on online algorithms in general. 1
2009 21st Euromicro Conference on RealTime Systems CompetitiveAnalysisofEnergyConstrainedRealTimeScheduling ∗
"... In this paper, we undertake the competitive analysis of the online realtime scheduling problems under a given hard energy constraint. Specifically, we derive worstcase performance bounds that apply to any online algorithm, when compared to an optimal algorithm that has the knowledge of the input s ..."
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In this paper, we undertake the competitive analysis of the online realtime scheduling problems under a given hard energy constraint. Specifically, we derive worstcase performance bounds that apply to any online algorithm, when compared to an optimal algorithm that has the knowledge of the input sequence in advance. First, by focusing on uniform valuedensity settings, we prove that no online algorithm can achieve a competitive factor greater than 1 − emax, where E emax is the upper bound on the size of any job and E is the available energy budget. Then we propose a variant ofEDF algorithm,ECEDF, that is able to achieve this upper bound. We show that a priori information about the largest job size in theactual inputsequence makes possible the design of asemionlinealgorithm ECEDF ∗ which achieves a constant competitive factor of 0.5. This turns out to be the best achievable competitive factor in these settings. We also extend our analysis to other settings, including those with nonuniform value densities and Dynamic Voltage Scaling capability. 1
Online Ranking of Split Graphs
"... A vertex ranking of a graph G is an assignment of positive integers (colors) to the vertices of G such that each path connecting two vertices of the same color contains a vertex of a higher color. Our main goal is to find a vertex ranking using as few colors as possible. Considering online algorith ..."
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A vertex ranking of a graph G is an assignment of positive integers (colors) to the vertices of G such that each path connecting two vertices of the same color contains a vertex of a higher color. Our main goal is to find a vertex ranking using as few colors as possible. Considering online algorithms for vertex ranking of split graphs, we prove that the worst case ratio of the number of colors used by any online ranking algorithm and the number of colors used in an optimal offline solution may be arbitrarily large. This negative result motivates us to investigate semi online algorithms, where a split graph is presented online but its clique number is given in advance. We prove that there does not exist a (2−ε)competitive semi online algorithm of this type. Finally, a 2competitive semi online algorithm is given.