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Nested Convex Bodies are Chaseable

Abstract

In the Convex Body Chasing problem, we are given an initial point \(v_0 \in \mathbb {R}^d\) and an online sequence of n convex bodies \(F_1, \ldots , F_n\). When we receive \(F_t\), we are required to move inside \(F_t\). Our goal is to minimize the total distance traveled. This fundamental online problem was first studied by Friedman and Linial (DCG 1993). They proved an \(\varOmega (\sqrt{d})\) lower bound on the competitive ratio, and conjectured that a competitive ratio depending only on d is possible. However, despite much interest in the problem, the conjecture remains wide open. We consider the setting in which the convex bodies are nested: \(F_1 \supset \cdots \supset F_n\). The nested setting is closely related to extending the online LP framework of Buchbinder and Naor (ESA 2005) to arbitrary linear constraints. Moreover, this setting retains much of the difficulty of the general setting and captures an essential obstacle in resolving Friedman and Linial’s conjecture. In this work, we give a f(d)-competitive algorithm for chasing nested convex bodies in \(\mathbb {R}^d\).

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Notes

  1. 1.

    If F is the intersection of halfspaces \(H_1, \ldots , H_s\), to simulate the request for F, the adversary can give \(H_1,\ldots ,H_s\) several times in a round-robin manner until the online algorithm moves inside F. Not revealing F directly can only hurt the online algorithm and does not affect the offline solution.

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Acknowledgements

We would like to thank Sébastien Bubeck, Niv Buchbinder, Anupam Gupta, Guru Guruganesh, Cristóbal Guzmán, and René Sitters for several interesting discussions.

Author information

Correspondence to Seeun William Umboh.

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This work was supported by NWO grant 639.022.211, ERC consolidator grant 617951, and GAČR grant P202/12/G061. A preliminary version of this paper appeared in SODA 2018 [7]. N. Bansal, M. Eliáš, G. Koumoutsos and S. W. Umboh: Research was done while the authors were at TU Eindhoven. M. Böhm: Research was done while the author was visiting TU Eindhoven.

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Bansal, N., Böhm, M., Eliáš, M. et al. Nested Convex Bodies are Chaseable. Algorithmica (2019). https://doi.org/10.1007/s00453-019-00661-x

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Keywords

  • Convex body chasing
  • Nested convex body chasing
  • Online algorithms
  • Competitive analysis