Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as : One is given a ground set @math , a collection @math of subsets thereof such that @math , and an objective (profit) function @math . The task is to choose a set @math that maximizes @math . We consider the version (, , both ICALP 2014) of such problems: The profit function @math (and possibly the set of feasible solutions @math ) may change over time. Since in many applications changing the solution is costly, the task becomes to find a sequence of solutions that optimizes the trade-off between good per-time solutions and stable solutions taking into account an additional similarity bonus. As similarity measure for two consecutive solutions, we consider either the size of the intersection of the two solutions or the difference of @math and the Hamming distance between the two characteristic vectors. We study multistage subset maximization problems in the setting, that is, @math (along with possibly @math ) only arrive one by one and, upon such an arrival, the online algorithm has to output the corresponding solution without knowledge of the future. We develop general techniques for online multistage subset maximization and thereby characterize those models (given by the type of data evolution and the type of similarity measure) that admit a constant-competitive online algorithm. When no constant competitive ratio is possible, we employ lookahead to circumvent this issue. When a constant competitive ratio is possible, we provide almost matching lower and upper bounds on the best achievable one.
The Multistage Max-Min Fair Allocation problem has been studied in the offline and the online settings in @cite_12 . This problem corresponds to a multistage variant of the Santa Klaus problem. For the off-line setting, the authors showed that the multistage version of the problem is much harder than the static one. They provided constant factor approximation algorithms for the off-line setting. For the online setting they proposed a constant competitive ratio for SSFS-type evolving instances and they proved that it is not possible to find an online algorithm with bounded competitive ratio for GE-type evolving instances. Finally, they showed that in the 1-lookahead case, where at time step @math we know the instance of time step @math , it is possible to get a constant approximation ratio.
@cite_12
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We consider the over-time version of the Max-Min Fair Allocation problem. Given a time horizon @math , with at each time t a set of demands and a set of available resources that may change over the time defining instance @math , we seek a sequence of solutions @math that (1) are near-optimal at each time t , and (2) as stable as possible (inducing small modification costs). We focus on the impact of the knowledge of the future on the quality and the stability of the returned solutions by distinguishing three settings: the off-line setting where the whole set of instances through the time horizon is known in advance, the on-line setting where no future instance is known, and the k -lookahead setting where at time t , the instances at times @math are known. We first consider the case without restrictions where the set of resources and the set of agents are the same for all instances and where every resource can be allocated to any agent. For the off-line setting, we show that the over-time version of the problem is much harder than the static one, since it becomes @math -hard even for families of instances for which the static problem is trivial. Then, we provide a @math -approximation algorithm for the off-line setting using as subroutine a -approximation algorithm for the static version. We also give a @math -competitive algorithm for the online setting using also as subroutine a -approximation algorithm for the static version. Furthermore, for the case with restrictions, we show that in the off-line setting it is possible to get a polynomial-time algorithm with the same approximation ratio as in the case without restrictions. For the online setting, we prove that it is not possible