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Competitively Chasing Convex Bodies
[article]

2018
*
arXiv
*
pre-print

In 1991, Linial and Friedman conjectured that the family of

arXiv:1811.00887v1
fatcat:q6aonfedmnbylp72c6v56qahwy
*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*. ...##
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Chasing Convex Bodies with Linear Competitive Ratio
[article]

2020
*
arXiv
*
pre-print

We study the problem of

arXiv:1905.11877v2
fatcat:ulh2pluglfcb7edtgp5lmyumme
*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. ...##
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Lipschitz Selectors may not Yield Competitive Algorithms for Convex Body Chasing
[article]

2022
*
arXiv
*
pre-print

It is natural to ask whether every selector with this Lipschitz property yields a

arXiv:2104.07487v2
fatcat:m2h6gotrdjbbbkyl4cdsslb4n4
*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 ...##
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Chasing Convex Bodies and Functions
[chapter]

2016
*
Lecture Notes in Computer Science
*

We consider three related online problems: Online

doi:10.1007/978-3-662-49529-2_6
fatcat:muwxki5lpzh2nkoywltpzbzo5q
*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*. ...##
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Nested Convex Bodies are Chaseable
[article]

2017
*
arXiv
*
pre-print

In this work, we give the first f(d)-

arXiv:1707.05527v1
fatcat:b22pz2ekdjfwzbxbilqwzbbyna
*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. ...##
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Chasing Convex Bodies Optimally
[article]

2021
*
arXiv
*
pre-print

The existence of a finite

arXiv:1905.11968v3
fatcat:fs55kpbd2fdedg4z3zv4djp7bq
*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. ...##
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Nested Convex Bodies are Chaseable
[chapter]

2018
*
Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms
*

In this work, we give a f (d)

doi:10.1137/1.9781611975031.81
dblp:conf/soda/BansalB0KU18
fatcat:qqg563nc55dshb5lvyz5rbdnum
*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*. ...##
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A Nearly-Linear Bound for Chasing Nested Convex Bodies
[article]

2018
*
arXiv
*
pre-print

Friedman and Linial introduced the

arXiv:1806.08865v2
fatcat:twphgb7xsnaddmcq6upnflybv4
*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. ...##
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Chasing Nested Convex Bodies Nearly Optimally
[article]

2021
*
arXiv
*
pre-print

The

arXiv:1811.00999v4
fatcat:3horsjgknrfeljkjfpbemgtaxa
*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*. ...##
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Online Multiserver Convex Chasing and Optimization
[article]

2020
*
arXiv
*
pre-print

We introduce the problem of k-

arXiv:2004.07346v1
fatcat:2qb6b3f3djbsta7j3njlovq4ly
*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 ...##
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Better Bounds for Online Line Chasing

2019
*
International Symposium on Mathematical Foundations of Computer Science
*

The line

doi:10.4230/lipics.mfcs.2019.8
dblp:conf/mfcs/BienkowskiBCCJK19
fatcat:lawdoulj65gydhonuziwl235vq
*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 ...##
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Better Bounds for Online Line Chasing
[article]

2019
*
arXiv
*
pre-print

The line

arXiv:1811.09233v2
fatcat:jbjth7w3vrhljlkwn3sbavtmb4
*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 ...##
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On convex body chasing

1993
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Discrete & Computational Geometry
*

We provide a strategy for the player which is

doi:10.1007/bf02189324
fatcat:s4kpcbhydbezzd7xzujzneymka
*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*. ...##
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Dimension-Free Bounds on Chasing Convex Functions
[article]

2020
*
arXiv
*
pre-print

We consider the problem of

arXiv:2005.14058v1
fatcat:22twcsug55f6fgj5hrgzjdr2qa
*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. ...##
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Green Computing Algorithmics

2011
*
2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
*

Most of the upper bounds in the literature for

doi:10.1109/focs.2011.44
dblp:conf/focs/Pruhs11
fatcat:6j6pfim4fndfvphbnfbornipcm
*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*. ...
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