scholarly journals Chasing Convex Bodies with Linear Competitive Ratio

Author(s):  
C.J. Argue ◽  
Anupam Gupta ◽  
Guru Guruganesh ◽  
Ziye Tang
2021 ◽  
Vol 68 (5) ◽  
pp. 1-10
Author(s):  
C. J. Argue ◽  
Anupam Gupta ◽  
Ziye Tang ◽  
Guru Guruganesh

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 .


1983 ◽  
Vol 48 (1) ◽  
pp. 192-198 ◽  
Author(s):  
Tomáš Boublík

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.


2020 ◽  
Vol 26 (1) ◽  
pp. 67-77 ◽  
Author(s):  
Silvestru Sever Dragomir

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}.


Algorithmica ◽  
2021 ◽  
Author(s):  
Matthias Englert ◽  
David Mezlaf ◽  
Matthias Westermann

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.


2002 ◽  
Vol 34 (06) ◽  
pp. 703-707 ◽  
Author(s):  
A. GIANNOPOULOS ◽  
M. HARTZOULAKI
Keyword(s):  

1997 ◽  
Vol 63 (4) ◽  
pp. 183-188
Author(s):  
Kunsoo Park ◽  
Sang Lyul Min ◽  
Yookun Cho

1964 ◽  
Vol 2 (2) ◽  
pp. 71-80 ◽  
Author(s):  
Nicolaas H. Kuiper
Keyword(s):  

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