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In this paper we introduce the notion of smoothed competitive analysis of online algorithms. Smoothed analysis has been proposed by [{\sl Spielman and Teng} STOC 2001] to explain the behaviour of algorithms that work well in practice while performing very poorly from a worst case analysis point of view. We apply this notion to analyze the Multi-Level Feedback (MLF) algorithm to minimize the total flow time on a sequence of jobs released over time when the processing time of a job is only known at time of completion. The initial processing times are integers in the range $[1,2^K]$. We use a partial bit randomization model, where the initial processing times are smoothened by changing the $k$ least significant bits under a quite general class of probability distributions. We show that MLF admits a smoothed competitive ratio of $O(max((2^k/\sigma)^3, (2^k/\sigma)^2 2^K-k))$, where $\sigma$ denotes the standard deviation of the distribution. In particular, we obtain a competitive ratio of $O(2^K-k)$ if $\sigma = \Theta(2^k)$. %The analysis holds for an oblivious as well as for a stronger adaptive %adversary. We also prove an $\Omega(2^{K-k})$ lower bound for any deterministic algorithm that is run on processing times smoothened according to the partial bit randomization model. For various other smoothening models, including the additive symmetric smoothening model used by [{\sl Spielman and Teng}], we give a higher lower bound of $\Omega(2^K)$. A direct consequence of our result is also the first average case analysis of MLF. We show a constant expected ratio of the total flow time of MLF to the optimum under several distributions including the uniform distribution.
Many online problems encountered in real-life involve a two-stage decision process: upon arrival of a new request, an irrevocable first-stage decision (the assignment of a specific resource to the request) must be made immediately, while in a second stage process, certain ``subinstances'' (that is, the instances of all requests assigned to a particular resource) can be solved to optimality (offline) later. We introduce the novel concept of an \emph{Online Target Date Assignment Problem} (\textsc{OnlineTDAP}) as a general framework for online problems with this nature. Requests for the \textsc{OnlineTDAP} become known at certain dates. An online algorithm has to assign a target date to each request, specifying on which date the request should be processed (e.\,g., an appointment with a customer for a washing machine repair). The cost at a target date is given by the \emph{downstream cost}, the optimal cost of processing all requests at that date w.\,r.\,t.\ some fixed downstream offline optimization problem (e.\,g., the cost of an optimal dispatch for service technicians). We provide general competitive algorithms for the \textsc{OnlineTDAP} independently of the particular downstream problem, when the overall objective is to minimize either the sum or the maximum of all downstream costs. As the first basic examples, we analyze the competitive ratios of our algorithms for the par ticular academic downstream problems of bin-packing, nonpreemptive scheduling on identical parallel machines, and routing a traveling salesman.