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Competitively Chasing Convex Bodies [article]

Sébastien Bubeck and Yin Tat Lee and Yuanzhi Li and Mark Sellke
2018 arXiv   pre-print
In 1991, Linial and Friedman conjectured that the family of convex sets in Euclidean space is chaseable. We prove this conjecture.  ...  In the F-chasing problem, an online algorithm observes a request sequence of sets in F and responds (online) by giving a sequence of points in these sets.  ...  Combining both points we would obtain a O(d 2 log d)-competitive algorithm for nested convex body chasing.  ... 
arXiv:1811.00887v1 fatcat:q6aonfedmnbylp72c6v56qahwy

Chasing Convex Bodies with Linear Competitive Ratio [article]

C.J. Argue, Anupam Gupta, Guru Guruganesh, Ziye Tang
2020 arXiv   pre-print
We study the problem of chasing convex bodies online: given a sequence of convex bodies K_t⊆R^d the algorithm must respond with points x_t∈ K_t in an online fashion (i.e., x_t is chosen before K_t+1 is  ...  (STOC 2019) gave a 2^O(d)-competitive algorithm for this problem. We give an algorithm that is O(min(d, √(d log T)))-competitive for any sequence of length T.  ...  There is an O(min(d, √ d log T ))-competitive algorithm for the general convex body chasing problem in d-dimensional Euclidean space.  ... 
arXiv:1905.11877v2 fatcat:ulh2pluglfcb7edtgp5lmyumme

Lipschitz Selectors may not Yield Competitive Algorithms for Convex Body Chasing [article]

C.J. Argue, Anupam Gupta, Marco Molinaro
2022 arXiv   pre-print
It is natural to ask whether every selector with this Lipschitz property yields a competitive algorithm for nested convex body chasing.  ...  The current best algorithms for convex body chasing problem in online algorithms use the notion of the Steiner point of a convex set.  ...  Introduction In the convex body chasing (CBC) problem, the player receives a sequence of nonempty closed convex sets K 1 , K 2 , . . . , K T ⊆ R d and must respond to each set K t with a point x t ∈ K  ... 
arXiv:2104.07487v2 fatcat:m2h6gotrdjbbbkyl4cdsslb4n4

Chasing Convex Bodies and Functions [chapter]

Antonios Antoniadis, Neal Barcelo, Michael Nugent, Kirk Pruhs, Kevin Schewior, Michele Scquizzato
2016 Lecture Notes in Computer Science  
We consider three related online problems: Online Convex Optimization, Convex Body Chasing, and Lazy Convex Body Chasing.  ...  And Convex Body Chasing is a special case of Lazy Convex Body Chasing where the destination point has to be in the convex region.  ...  implies a c-competitive algorithm for Convex Body Chasing.  ... 
doi:10.1007/978-3-662-49529-2_6 fatcat:muwxki5lpzh2nkoywltpzbzo5q

Nested Convex Bodies are Chaseable [article]

Nikhil Bansal, Martin Böhm, Marek Eliáš, Grigorios Koumoutsos, Seeun William Umboh
2017 arXiv   pre-print
In this work, we give the first f(d)-competitive algorithm for chasing nested convex bodies in R^d.  ...  In the Convex Body Chasing problem, we are given an initial point v_0 in R^d and an online sequence of n convex bodies F_1, ..., F_n. When we receive F_i, we are required to move inside F_i.  ...  (d)-competitive for convex body chasing. Recently, Antoniadis et al.  ... 
arXiv:1707.05527v1 fatcat:b22pz2ekdjfwzbxbilqwzbbyna

Chasing Convex Bodies Optimally [article]

Mark Sellke
2021 arXiv   pre-print
The existence of a finite competitive ratio for convex body chasing was first conjectured in 1991 by Friedman and Linial.  ...  In the chasing convex bodies problem, an online player receives a request sequence of N convex sets K_1,..., K_N contained in a normed space ℝ^d.  ...  Acknowledgement I thank Sébastien Bubeck, Bo'az Klartag, Yin Tat Lee, and Yuanzhi Li for the introduction to convex body chasing and the Steiner point, and many stimulating discussions.  ... 
arXiv:1905.11968v3 fatcat:fs55kpbd2fdedg4z3zv4djp7bq

Nested Convex Bodies are Chaseable [chapter]

Nikhil Bansa, Martin Böhm, Marek Eliáš, Grigorios Koumoutsos, Seeun William Umboh
2018 Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms  
In this work, we give a f (d)competitive algorithm for chasing nested convex bodies in R d .  ...  In the Convex Body Chasing problem, we are given an initial point v 0 ∈ R d and an online sequence of n convex bodies F 1 , . . . , F n . When we receive F i , we are required to move inside F i .  ...  nearly-optimal competitive ratios for many MSS and is a natural candidate to be f (d)-competitive for convex body chasing.  ... 
doi:10.1137/1.9781611975031.81 dblp:conf/soda/BansalB0KU18 fatcat:qqg563nc55dshb5lvyz5rbdnum

A Nearly-Linear Bound for Chasing Nested Convex Bodies [article]

C.J. Argue and Sébastien Bubeck and Michael B. Cohen and Anupam Gupta and Yin Tat Lee
2018 arXiv   pre-print
Friedman and Linial introduced the convex body chasing problem to explore the interplay between geometry and competitive ratio in metrical task systems.  ...  In convex body chasing, at each time step t ∈N, the online algorithm receives a request in the form of a convex body K_t ⊆R^d and must output a point x_t ∈ K_t.  ...  Introduction We consider the convex body chasing problem.  ... 
arXiv:1806.08865v2 fatcat:twphgb7xsnaddmcq6upnflybv4

Chasing Nested Convex Bodies Nearly Optimally [article]

Sébastien Bubeck, Bo'az Klartag, Yin Tat Lee, Yuanzhi Li, Mark Sellke
2021 arXiv   pre-print
The convex body chasing problem, introduced by Friedman and Linial, is a competitive analysis problem on any normed vector space.  ...  In convex body chasing, for each timestep t∈ℕ, a convex body K_t⊆ℝ^d is given as a request, and the player picks a point x_t∈ K_t.  ...  Our companion paper [BLLS18] establishes the first finite upper bound for the competitive ratio of convex body chasing.  ... 
arXiv:1811.00999v4 fatcat:3horsjgknrfeljkjfpbemgtaxa

Online Multiserver Convex Chasing and Optimization [article]

Sébastien Bubeck, Yuval Rabani, Mark Sellke
2020 arXiv   pre-print
We introduce the problem of k-chasing of convex functions, a simultaneous generalization of both the famous k-server problem in R^d, and of the problem of chasing convex bodies and functions.  ...  with dimension-free competitive ratio.  ...  Chasing convex functions online (by a single server) is also a problem with a long history, starting with the convex body chasing problem of [18] (a chased convex body can be viewed as a convex function  ... 
arXiv:2004.07346v1 fatcat:2qb6b3f3djbsta7j3njlovq4ly

Better Bounds for Online Line Chasing

Marcin Bienkowski, Jaroslaw Byrka, Marek Chrobak, Christian Coester, Lukasz Jez, Elias Koutsoupias, Michael Wagner
2019 International Symposium on Mathematical Foundations of Computer Science  
The line chasing problem is a variant of a more general convex body chasing problem, where the sets Xt are arbitrary convex sets.  ...  The latter bound also improves upon the previous lower bound of √ 2 ≈ 1.412 for convex body chasing in 2 dimensions.  ...  The convex body chasing problem was originally introduced in 1993 by Friedman and Linial [11] , who gave a constant-competitive algorithm for chasing convex bodies in R 2 (the plane) and conjectured that  ... 
doi:10.4230/lipics.mfcs.2019.8 dblp:conf/mfcs/BienkowskiBCCJK19 fatcat:lawdoulj65gydhonuziwl235vq

Better Bounds for Online Line Chasing [article]

Marcin Bienkowski, Jarosław Byrka, Marek Chrobak, Christian Coester, Łukasz Jeż, Elias Koutsoupias
2019 arXiv   pre-print
The line chasing problem is a variant of a more general convex body chasing problem, where the sets X_t are arbitrary convex sets.  ...  To date, the best competitive ratio for the line chasing problem was 28.1, even in the plane. We significantly improve this bound, by providing a 3-competitive algorithm for any dimension d.  ...  The convex body chasing problem was originally introduced in 1993 by Friedman and Linial [11] , who gave a constant-competitive algorithm for chasing convex bodies in R 2 (the plane) and conjectured that  ... 
arXiv:1811.09233v2 fatcat:jbjth7w3vrhljlkwn3sbavtmb4

On convex body chasing

Joel Friedman, Nathan Linial
1993 Discrete & Computational Geometry  
We provide a strategy for the player which is competitive, i.e., for any sequence Fi the cost to the player is within a constant (multiplicative) factor of the "off-line" cost (i.e., the least possible  ...  A player moving in the plane is given a sequence of instructions of the following type: at step i a planar convex set Fi is specified, and the player has to move to a point in Fi.  ...  Line Chasing in R" Although we do not know whether or not convex bodies in R 3 can be chased, we can say that lines in 113 can be competitively chased.  ... 
doi:10.1007/bf02189324 fatcat:s4kpcbhydbezzd7xzujzneymka

Dimension-Free Bounds on Chasing Convex Functions [article]

C.J. Argue, Anupam Gupta, Guru Guruganesh
2020 arXiv   pre-print
We consider the problem of chasing convex functions, where functions arrive over time.  ...  In particular, we consider the case where the convex functions are κ-well-conditioned, and give an algorithm that achieves an O(√(κ))-competitiveness.  ...  Suppose there is an g(d)-competitive algorithm for chasing convex bodies in R d , for each d ≥ 1.  ... 
arXiv:2005.14058v1 fatcat:22twcsug55f6fgj5hrgzjdr2qa

Green Computing Algorithmics

Kirk Pruhs
2011 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science  
Most of the upper bounds in the literature for Convex Body Chasing are for chasing certain special types of convex bodies.  ...  Open Problem: Find a provably O(1)-competitive algorithm for the special case of Convex Body Chasing.  ... 
doi:10.1109/focs.2011.44 dblp:conf/focs/Pruhs11 fatcat:6j6pfim4fndfvphbnfbornipcm
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