Chasing Convex Bodies with Linear Competitive Ratio

Journal of the ACM ◽

10.1145/3450349 ◽

2021 ◽

Vol 68 (5) ◽

pp. 1-10

Author(s):

C. J. Argue ◽

Anupam Gupta ◽

Ziye Tang ◽

Guru Guruganesh

Keyword(s):

Competitive Ratio ◽

Convex Bodies ◽

Competitive Algorithm ◽

Sum Of Distances

We study the problem of chasing convex bodies online: given a sequence of convex bodies the algorithm must respond with points in an online fashion (i.e., is chosen before is revealed). The objective is to minimize the sum of distances between successive points in this sequence. Bubeck et al. (STOC 2019) gave a -competitive algorithm for this problem. We give an algorithm that is -competitive for any sequence of length .

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Chasing convex bodies with linear competitive ratio (invited paper)

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing ◽

10.1145/3406325.3465354 ◽

2021 ◽

Author(s):

C. J. Argue ◽

Anupam Gupta ◽

Guru Guruganesh ◽

Ziye Tang

Keyword(s):

Competitive Ratio ◽

Convex Bodies

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Convex Bodies: The Brunn–Minkowski Theory

10.1017/cbo9780511526282 ◽

1993 ◽

Cited By ~ 891

Author(s):

Rolf Schneider

Keyword(s):

Convex Bodies

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Excess entropy of the systems of molecules differing in size and shape

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19830192 ◽

1983 ◽

Vol 48 (1) ◽

pp. 192-198 ◽

Cited By ~ 5

Author(s):

Tomáš Boublík

Keyword(s):

Equation Of State ◽

Hard Spheres ◽

Excess Entropy ◽

Convex Bodies ◽

Pure Substance ◽

Liquid Equilibrium ◽

Simple Rules ◽

Different Shapes ◽

The Mean

The excess entropy of mixing of mixtures of hard spheres and spherocylinders is determined from an equation of state of hard convex bodies. The obtained dependence of excess entropy on composition was used to find the accuracy of determining ΔSE from relations employed for the correlation and prediction of vapour-liquid equilibrium. Simple rules were proposed for establishing the mean parameter of nonsphericity for mixtures of hard bodies of different shapes allowing to describe the P-V-T behaviour of solutions in terms of the equation of state fo pure substance. The determination of ΔSE by means of these rules is discussed.

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Some Hermite–Hadamard type integral inequalities for convex functions defined on convex bodies in ℝn\mathbb{R}^{n}

Journal of Applied Analysis ◽

10.1515/jaa-2020-2005 ◽

2020 ◽

Vol 26 (1) ◽

pp. 67-77 ◽

Cited By ~ 1

Author(s):

Silvestru Sever Dragomir

Keyword(s):

Euclidean Space ◽

Convex Functions ◽

Convex Bodies ◽

Integral Inequalities ◽

Divergence Theorem ◽

Several Variables ◽

Type Integral ◽

Bounded Convex

AbstractIn this paper, by the use of the divergence theorem, we establish some integral inequalities of Hermite–Hadamard type for convex functions of several variables defined on closed and bounded convex bodies in the Euclidean space {\mathbb{R}^{n}} for any {n\geq 2}.

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Online Makespan Scheduling with Job Migration on Uniform Machines

Algorithmica ◽

10.1007/s00453-021-00852-5 ◽

2021 ◽

Author(s):

Matthias Englert ◽

David Mezlaf ◽

Matthias Westermann

Keyword(s):

Competitive Ratio ◽

Online Algorithm ◽

Parallel Machines ◽

Scheduling Algorithm ◽

Input Sequence ◽

Job Migration ◽

Average Completion Time ◽

Uniform Machine ◽

Uniform Machines ◽

Completion Times

AbstractIn the classic minimum makespan scheduling problem, we are given an input sequence of n jobs with sizes. A scheduling algorithm has to assign the jobs to m parallel machines. The objective is to minimize the makespan, which is the time it takes until all jobs are processed. In this paper, we consider online scheduling algorithms without preemption. However, we allow the online algorithm to change the assignment of up to k jobs at the end for some limited number k. For m identical machines, Albers and Hellwig (Algorithmica 79(2):598–623, 2017) give tight bounds on the competitive ratio in this model. The precise ratio depends on, and increases with, m. It lies between 4/3 and $$\approx 1.4659$$ ≈ 1.4659 . They show that $$k = O(m)$$ k = O ( m ) is sufficient to achieve this bound and no $$k = o(n)$$ k = o ( n ) can result in a better bound. We study m uniform machines, i.e., machines with different speeds, and show that this setting is strictly harder. For sufficiently large m, there is a $$\delta = \varTheta (1)$$ δ = Θ ( 1 ) such that, for m machines with only two different machine speeds, no online algorithm can achieve a competitive ratio of less than $$1.4659 + \delta $$ 1.4659 + δ with $$k = o(n)$$ k = o ( n ) . We present a new algorithm for the uniform machine setting. Depending on the speeds of the machines, our scheduling algorithm achieves a competitive ratio that lies between 4/3 and $$\approx 1.7992$$ ≈ 1.7992 with $$k = O(m)$$ k = O ( m ) . We also show that $$k = \varOmega (m)$$ k = Ω ( m ) is necessary to achieve a competitive ratio below 2. Our algorithm is based on maintaining a specific imbalance with respect to the completion times of the machines, complemented by a bicriteria approximation algorithm that minimizes the makespan and maximizes the average completion time for certain sets of machines.

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