Achieving Competitiveness in Online Problems
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In the setting of online algorithms, the input is initially not present but rather arrive one-by-one over time and after each input, the algorithm has to make a decision. Depending on the formulation of the problem, the algorithm might be allowed to change its previous decisions or not at a later time. We analyze two problems to show that it is possible for an online algorithm to become more competitive by changing its former decisions. We first consider the online edge orientation in which the edges arrive one-by-one to an empty graph and the aim is to orient them in a way such that the maximum in-degree is minimized. We then consider the online bipartite b-matching. In this problem, we are given a bipartite graph where one side of the graph is initially present and where the other side arrive online. The goal is to maintain a matching set such that the maximum degree in the set is minimized. For both of the problems, the best achievable competitive ratio is Θ(log n) over n input arrivals when decisions are irreversible. We study three algorithms for these problems, two for the former and one for the latter, that achieve O(1) competitive ratio by changing O(n) of their decisions over n arrivals. In addition to that, we analyze one of the algorithms, the shortest path algorithm, against an adversary. Through that, we prove some new results about algorithms performance.
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