International audienceIn this paper, we study the advice complexity of the online bin packing problem. In this well-studied setting, the online algorithm is supplemented with some additional information concerning the input. We improve upon both known upper and lower bounds of online algorithms for this problem. On the positive side, we first provide a relatively simple algorithm that achieves a competitive ratio arbitrarily close to 1.5, using constant-size advice. Our result implies that 16 bits of advice suffice to obtain a competitive ratio better than any online algorithm without advice, thus improving the previously known bound of O(log(n)) bits required to attain this performance. In addition, we introduce a more complex algorithm that still requires only constant-size advice, and which is below 1.5-competitive, namely has competitive ratio arbitrarily close to 1.47012. This is the currently best performance of any online bin packing algorithm with sublinear advice. On the negative side, we extend a construction due to Boyar et al. [10] so as to show that no online algorithm with sub-linear advice can be 7/6-competitive, which improves upon the known lower bound of 9/8

It has long been known that for the paging problem in its standard form, competitive analysis cannot adequately distinguish algorithms based on their performance: there exists a vast class of algorithms which achieve the same competitive ratio, ranging from extremely naive and inefficient strategies (such as Flush-When-Full), to strategies of excellent performance in practice (such as Least-Recently-Used and some of its variants). A similar situation arises in the list update problem: in particular, under the cost formulation studied by Martínez and Roura [TCS 2000] and Munro [ESA 2000] every list update algorithm has, asymptotically, the same competitive ratio. Several refinements of competitive analysis, as well as alternative performance measures have been introduced in the literature, with varying degrees of success in narrowing this disconnect between theoretical analysis and empirical evaluation.In this paper we study these two fundamental online problems under the framework of bijective analysis [Angelopoulos, López-Ortiz, SODA 2007 and LATIN 2008]. This is an intuitive technique which is based on pairwise comparison of the costs incurred by two algorithms on sets of request sequences of the same size. Coupled with a well-established model of locality of reference due to Albers, Favrholdt and Giel [JCSS 2005], we show that LeastRecently-Used and Move-to-Front are the unique optimal algorithms for paging and list update, respectively. Prior to this work, only measures based on average-cost analysis have separated LRU and MTF from all other algorithms. Given that bijective analysis is a fairly stringent measure (and also subsumes average-cost analysis), we prove that in a strong sense LRU and MTF stand out as the best (deterministic) algorithms.

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We introduce a new technique for the analysis of online algorithms, namely bijective analysis, that is based on pair-wise comparison of the costs incurred by the algorithms. Under this framework, an algorithm A is no worse than an algorithm B if there is a bijection π defined over all request sequences of a given size such that the cost of A on σ is no more than the cost of B on B(π(σ)). We also study a relaxation of bijective analysis, termed average analysis, in which we compare two algorithms based on their corresponding average costs over request sequences of a given size. We apply these new techniques in the context of two fundamental online problems, namely paging and list update. For paging, we show that any two lazy online algorithms are equivalent under bijective analysis. This result demonstrates that, without further assumptions on characteristics of request sequences, it is unlikely, or even undesirable, to separate online paging algorithms based on their performance. However, once we restrict the set of request sequences to those exhibiting locality of reference, and in particular using a model of locality due to Albers, Favrholdt, and Giel [JCSS 2005], we demonstrate that Least-Recently-Used (LRU) is the unique optimal strategy according to average analysis. This is, to our knowledge, the first deterministic model to provide full theoretical backing to the empirical observation that LRU is preferable in practice. Concerning list update, we obtain similar conclusions, in terms of the bijective comparison of any two online algorithms, and in terms of the superiority (albeit not necessarily unique) of the Move-To-Front (MTF) heuristic in the presence of locality of reference.

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The edge asymmetry of a directed, edge-weighted graph is defined as the maximum ratio of the weight of antiparallel edges in the graph, and can be used as a measure of the heterogeneity of links in a data communication network. In this paper we provide a near-tight upper bound on the competitive ratio of the Online Steiner Tree problem in graphs of bounded edge asymmetry α. This problem has applications in efficient multicasting over networks with non-symmetric links. We show an improved upper bound of O min max α log k log α , α log k log log k , k on the competitive ratio of a simple greedy algorithm, for any request sequence of k terminals. The result almost matches the lower bound of Ω min max α log k log α , α log k log log k , k 1− (where is an arbitrarily small constant) due to Faloutsos et al.[8] and Angelopoulos [3].

Borodin, Nielsen and Rackoff [13] introduced the class of priority algorithms as a framework for modeling deterministic greedy-like algorithms. In this paper we address the effect of randomization in greedy-like algorithms. More specifically, we consider approximation ratios within the context of randomized priority algorithms. As case studies, we prove inapproximation results for two well-studied optimization problems, namely facility location and makespan scheduling.

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