An on-line graph coloring algorithm with sublinear performance ratio
Discrete Mathematics
Lower bounds for on-line graph coloring
Theoretical Computer Science - Special issue on dynamic and on-line algorithms
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Average-Case Competitive Analyses for Ski-Rental Problems
ISAAC '02 Proceedings of the 13th International Symposium on Algorithms and Computation
Removable Online Knapsack Problems
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
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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 diffcult 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 diffculty 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 nk operations per vertex.