Online independent sets

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 dicult 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 diculty 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 rst, the algorithm can maintain a polynomial number of solutions (independent sets) and choose the largest one as the nal solution. We show that (formula presented) 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 dierent 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.