Abstract
At each step of the online independent set problem, we are given a vertex v and its edges to the previously given vertices. We are to decide whether or not to select v as a member of an independent set. Our goal is to maximize the size of the independent set. It is not dificult to see that no online algorithm can attain a performance ratio better than n - 1, where n denotes the total number of vertices. Given this extreme dificulty of the problem, we study here relaxations where the algorithm can hedge his bets by maintaining multiple alternative solutions simultaneously.
We introduce two models. In the first, the algorithm can maintain a polynomial number of solutions (independent sets) and choose the largest one as the final solution. We show that θ(n/log n) is the best competitive ratio for this model. In the second more powerful model, the algorithm can copy intermediate solutions and grow the copied solutions in different ways. We obtain an upper bound of O(n/(k log n)), and a lower bound of n/(ek+1 log3 n), when the algorithm can make n k operations per vertex.
Supported in part by Scientific Research Grant, Ministry of Japan, 10558044, 09480055 and 10205215.
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Halldórsson, M.M., Iwama, K., Miyazaki, S., Taketomi, S. (2000). Online Independent Sets. In: Du, DZ., Eades, P., Estivill-Castro, V., Lin, X., Sharma, A. (eds) Computing and Combinatorics. COCOON 2000. Lecture Notes in Computer Science, vol 1858. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44968-X_20
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DOI: https://doi.org/10.1007/3-540-44968-X_20
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