Abstract
We consider three related online problems: Online Convex Optimization, Convex Body Chasing, and Lazy Convex Body Chasing. In Online Convex Optimization the input is an online sequence of convex functions over some Euclidean space. In response to a function, the online algorithm can move to any destination point in the Euclidean space. The cost is the total distance moved plus the sum of the function costs at the destination points. Lazy Convex Body Chasing is a special case of Online Convex Optimization where the function is zero in some convex region, and grows linearly with the distance from this region. And Convex Body Chasing is a special case of Lazy Convex Body Chasing where the destination point has to be in the convex region. We show that these problems are equivalent in the sense that if any of these problems have an O(1)-competitive algorithm then all of the problems have an O(1)-competitive algorithm. By leveraging these results we then obtain the first O(1)-competitive algorithm for Online Convex Optimization in two dimensions, and give the first O(1)-competitive algorithm for chasing linear subspaces. We also give a simple algorithm and O(1)-competitiveness analysis for chasing lines.
K. Pruhs—Supported, in part, by NSF grants CCF-1115575, CNS-1253218, CCF-1421508, and an IBM Faculty Award.
K. Schewior—Supported by the DFG within the research training group ‘Methods for Discrete Structures’ (GRK 1408).
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Acknowledgement
We thank Nikhil Bansal, Anupam Gupta, Cliff Stein, Ravishankar Krishnaswamy, and Adam Wierman for helpful discussions. We also thank an anonymous reviewer for pointing out an important subtlety in one of our proofs.
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Antoniadis, A., Barcelo, N., Nugent, M., Pruhs, K., Schewior, K., Scquizzato, M. (2016). Chasing Convex Bodies and Functions. In: Kranakis, E., Navarro, G., Chávez, E. (eds) LATIN 2016: Theoretical Informatics. LATIN 2016. Lecture Notes in Computer Science(), vol 9644. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49529-2_6
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DOI: https://doi.org/10.1007/978-3-662-49529-2_6
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