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In online graph coloring a graph is revealed to an online algorithm one vertex at a time, and the algorithm must color the vertices as they appear. This paper starts to investigate the advice complexity of this problem-the amount of oracle information an online algorithm needs in order to make optimal choices. We also consider a more general problem-a trade-off between online and offline graph coloring. In the paper we prove that precisely n/2 − 1 bits of advice are needed when the vertices on a path are presented for coloring in arbitrary order. The same holds in the more general case when just a subset of the vertices is colored online. However, the problem turns out to be non-trivial for the case where the online algorithm is guaranteed that the vertices it receives form a subset of a path and are presented in the order in which they lie on the path. For this variant we prove that its advice complexity is βn + O(log n) bits, where β ≈ 0.406 is a fixed constant (we give its closed form). This suggests that the generalized problem will be challenging for more complex graph classes.

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Abstract. In this paper we show a polynomial-time algorithm to find the best rational approximation of a given rational number within a given interval. As a special case, we show how to find the best rational number that after evaluating and rounding exactly matches the input number. In both results, "best" means "having the smallest possible denominator".

The paper investigates a version of the resource allocation problem arising in the wireless networking, namely in the OVSF code reallocation process. In this setting a complete binary tree of a given height n is considered, together with a sequence of requests which have to be served in an online manner. The requests are of two types: an insertion request requires to allocate a complete subtree of a given height, and a deletion request frees a given allocated subtree. In order to serve an insertion request it might be necessary to move some already allocated subtrees to other locations in order to free a large enough subtree. We are interested in the worst case average number of such reallocations needed to serve a request.In [4] the authors delivered bounds on the competitive ratio of online algorithm solving this problem, and showed that the ratio is between 1.5 and O(n). We partially answer their question about the exact value by giving an O(1)-competitive online algorithm.In [3], authors use the same model in the context of memory management systems, and analyze the number of reallocations needed to serve a request in the worst case. In this setting, our result is a corresponding amortized analysis.Classification: Algorithms and data structures