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A note on the integrality gap of the configuration LP for restricted Santa Claus
[article]

2018
*
arXiv
*
pre-print

In

arXiv:1807.03626v1
fatcat:ymbla2q3tzdvlgkqscs7lmqhou
*the*original proof,*a*local search was used to obtain*a*bound*of*4*for**the*ratio*of**the*fractional to*the**integral*optimum*of**the**configuration**LP*(*integrality**gap*). ... In*the**restricted**Santa**Claus*problem we are given resources R and players P. Every resource j∈ R has*a*value v_j and every player i desires*a*set R(i)*of*resources. ...*For**the**restricted**Santa**Claus*problem there exists*a*strong*LP*relaxation,*the**configuration**LP*.*The*proof that this has*a*small*integrality**gap*(see [2] ) is not trivial. ...##
###
A polynomial time 12-approximation algorithm for restricted Santa Claus problem
[article]

2020
*
arXiv
*
pre-print

Our algorithm starts by solving

arXiv:2007.09849v2
fatcat:2iei7dt6zbc7xfs5rufzlfiwwe
*the**configuration**LP*and uses*the*optimum value to get*a*12-*gap*instance. This is then followed by*the*well-known clustering technique*of*Bansal and Sviridenko. ... In this paper, we consider*the**restricted*case*of**the*problem and improve*the*current best approximation ratio by presenting*a*polynomial time 12-approximation algorithm using linear programming and semi-definite ... This work is motivated by*the*existence*of*large*gap*between*the*best theoretical*integrality**gap**of*3*for**the**restricted**Santa**Claus*problem as evident from result due to Asadpour et al. ...##
###
Santa claus meets hypergraph matchings

2012
*
ACM Transactions on Algorithms
*

As

doi:10.1145/2229163.2229168
fatcat:juenfplt6nahljvxesvpsairja
*a*result, we prove that*the**integrality**gap**of**the**configuration**LP*is no worse than 1 4 . ... It was previously shown via*a*nonconstructive proof that uses*the*Lovász local lemma that*the**integrality**gap**of**a*certain*configuration**LP**for**the*problem is no worse than some (unspecified) constant. ... Acknowledgements Part*of*this work was performed at Microsoft Research, Redmond, Washington. ...##
###
Improved Integrality Gap in Max-Min Allocation: or Topology at the North Pole
[article]

2022
*
arXiv
*
pre-print

We study

arXiv:2202.01143v1
fatcat:6q3h5ywhwvf3xjrsx5o3lhmeaq
*the**restricted*variant, also known as*the**Santa**Claus*problem, where each resource has an intrinsic positive value, and each player covets*a*subset*of**the*resources. ...*The*principal approach*for*obtaining approximation algorithms has been via*the**Configuration**LP*(CLP)*of*Bansal and Sviridenko. ... Acknowledgements:*The*authors are grateful to Lothar Narins*for*helpful discussions in*the*early stages*of*this work, and to Olaf Parczyk*for*computing*the*data in Section 7. ...##
###
Quasi-polynomial Local Search for Restricted Max-Min Fair Allocation
[chapter]

2012
*
Lecture Notes in Computer Science
*

*The*

*restricted*max-min fair allocation problem (also known as

*the*

*restricted*

*Santa*

*Claus*problem) is

*one*

*of*few problems that enjoys

*the*intriguing status

*of*having

*a*better estimation algorithm than approximation ... Indeed, Asadpour et al. [1] proved that

*a*certain

*configuration*

*LP*can be used to estimate

*the*optimal value within

*a*factor 1/(4 + ),

*for*any > 0, but at

*the*same time it is not known how to efficiently ... This case is known as

*the*

*restricted*max-min fair allocation or

*the*

*restricted*

*Santa*

*Claus*problem and is

*the*focus

*of*our paper. ...

##
###
Santa Claus Schedules Jobs on Unrelated Machines
[article]

2011
*
arXiv
*
pre-print

*The*result is obtained by upper bounding

*the*

*integrality*

*gap*

*of*

*a*certain strong linear program, known as

*configuration*

*LP*, that was previously successfully used

*for*

*the*related

*Santa*

*Claus*problem. ...

*One*

*of*

*the*classic results in scheduling theory is

*the*2-approximation algorithm by Lenstra, Shmoys, and Tardos

*for*

*the*problem

*of*scheduling jobs to minimize makespan

*on*unrelated machines, i.e., job ... Acknowledgements I am grateful to Johan Håstad

*for*many useful insights and comments. I also wish to thank Tobias Mömke and Lukáš Poláček

*for*useful comments

*on*

*the*exposition. ...

##
###
Quasi-Polynomial Local Search for Restricted Max-Min Fair Allocation
[article]

2014
*
arXiv
*
pre-print

*The*

*restricted*max-min fair allocation problem (also known as

*the*

*restricted*

*Santa*

*Claus*problem) is

*one*

*of*few problems that enjoys

*the*intriguing status

*of*having

*a*better estimation algorithm than approximation ... Indeed, Asadpour et al. proved that

*a*certain

*configuration*

*LP*can be used to estimate

*the*optimal value within

*a*factor 1/(4+ϵ),

*for*any ϵ>0, but at

*the*same time it is not known how to efficiently find ... This case is known as

*the*

*restricted*max-min fair allocation or

*the*

*restricted*

*Santa*

*Claus*problem and is

*the*focus

*of*our paper. ...

##
###
Quasi-Polynomial Local Search for Restricted Max-Min Fair Allocation

2015
*
ACM Transactions on Algorithms
*

*The*

*restricted*max-min fair allocation problem (also known as

*the*

*restricted*

*Santa*

*Claus*problem) is

*one*

*of*few problems that enjoys

*the*intriguing status

*of*having

*a*better estimation algorithm than approximation ... Indeed, Asadpour et al. [1] proved that

*a*certain

*configuration*

*LP*can be used to estimate

*the*optimal value within

*a*factor 1/(4 + ),

*for*any > 0, but at

*the*same time it is not known how to efficiently ... This case is known as

*the*

*restricted*max-min fair allocation or

*the*

*restricted*

*Santa*

*Claus*problem and is

*the*focus

*of*our paper. ...

##
###
Santa Claus schedules jobs on unrelated machines

2011
*
Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11
*

*The*result is obtained by upper bounding

*the*

*integrality*

*gap*

*of*

*a*certain strong linear program, known as

*configuration*

*LP*, that was previously successfully used

*for*

*the*related

*Santa*

*Claus*problem. ...

*One*

*of*

*the*classic results in scheduling theory is

*the*2-approximation algorithm by Lenstra, Shmoys, and Tardos

*for*

*the*problem

*of*scheduling jobs to minimize makespan

*on*unrelated machines, i.e., job ... Acknowledgements I am grateful to Johan Håstad

*for*many useful insights and comments. I also wish to thank Tobias Mömke and Lukáš Poláček

*for*useful comments

*on*

*the*exposition. ...

##
###
Santa Claus Schedules Jobs on Unrelated Machines

2012
*
SIAM journal on computing (Print)
*

*The*result is obtained by upper bounding

*the*

*integrality*

*gap*

*of*

*a*certain strong linear program, known as

*configuration*

*LP*, that was previously successfully used

*for*

*the*related

*Santa*

*Claus*problem. ...

*One*

*of*

*the*classic results in scheduling theory is

*the*2-approximation algorithm by Lenstra, Shmoys, and Tardos

*for*

*the*problem

*of*scheduling jobs to minimize makespan

*on*unrelated machines, i.e., job ... Acknowledgements I am grateful to Johan Håstad

*for*many useful insights and comments. I also wish to thank Tobias Mömke and Lukáš Poláček

*for*useful comments

*on*

*the*exposition. ...

##
###
A Tale of Santa Claus, Hypergraphs and Matroids
[article]

2019
*
arXiv
*
pre-print

Our result can then be used as

arXiv:1807.07189v2
fatcat:i4nepjnh4ra6lb56av2zoc5aky
*a*blackbox to obtain*a*(4+ε)-approximation*for**Santa**Claus*. This factor also compares against*a*natural, compact*LP**for**Santa**Claus*. ... In this paper, we introduce*a*matroid version*of**the**Santa**Claus*problem. ... Interestingly,*the**integrality**gap**of**the**configuration**LP*is at least Ω( n) [BS06] . ...##
###
The Combinatorial Santa Claus Problem or: How to Find Good Matchings in Non-Uniform Hypergraphs
[article]

2020
*
arXiv
*
pre-print

*The*non-trivial observation that any uniform regular hypergraph admits

*a*relaxed matching

*for*α = O(1) was

*a*major step in obtaining

*a*constant approximation rate

*for*

*a*special case

*of*

*the*

*Santa*

*Claus*... We dub this problem

*the*Combinatorial

*Santa*

*Claus*problem, since we show in this paper that this problem and

*the*

*Santa*

*Claus*problem are almost equivalent in terms

*of*their approximability. ... Namely, it implies an O(α)

*integrality*

*gap*

*for*an

*LP*relaxation

*of*

*the*

*restricted*

*Santa*

*Claus*problem, which is

*the*special case where

*for*all valuations v ij ∈ {0, v j } (v j being

*a*player independent ...

##
###
The Submodular Santa Claus Problem in the Restricted Assignment Case
[article]

2020
*
arXiv
*
pre-print

*The*submodular

*Santa*

*Claus*problem was introduced in

*a*seminal work by Goemans, Harvey, Iwata, and Mirrokni (SODA'09) as an application

*of*their structural result. ... In particular,

*a*line

*of*research has shown that there is

*a*polynomial time constant approximation algorithm

*for*linear valuation functions in

*the*

*restricted*assignment case. ...

*A*

*note*

*on*

*the*

*integrality*

*gap*

*of*

*the*

*configuration*

*lp*

*for*

*restricted*

*santa*

*claus*. Information Processing Letters, 164:106025, 2020. [LLN06] Benny Lehmann, Daniel Lehmann, and Noam Nisan. ...

##
###
New Constructive Aspects of the Lovasz Local Lemma

2010
*
2010 IEEE 51st Annual Symposium on Foundations of Computer Science
*

We demonstrate this idea

doi:10.1109/focs.2010.45
dblp:conf/focs/HaeuplerSS10
fatcat:uaon4s3s5fdhjj5rhdddzlvnum
*on*several applications. We give*the*first constant-factor approximation algorithm*for**the**Santa**Claus*problem by making an LLL-based proof*of*Feige constructive. ... We show how*a*known bound*on**the*probabilities*of*events in this distribution can be used*for*further probabilistic analysis and give new constructive and non-constructive results. ...*The*first author thanks Michel Goemans and Ankur Moitra*for*discussing*the**Santa**Claus*problem with him in an early stage*of*this work. ...##
###
New Constructive Aspects of the Lovász Local Lemma

2011
*
Journal of the ACM
*

We demonstrate this idea

doi:10.1145/2049697.2049702
fatcat:yshjonds4bgkpcwrhhss3tdmhi
*on*several applications. We give*the*first constant-factor approximation algorithm*for**the**Santa**Claus*problem by making an LLL-based proof*of*Feige constructive. ... We show how*a*known bound*on**the*probabilities*of*events in this distribution can be used*for*further probabilistic analysis and give new constructive and non-constructive results. ...*The*first author thanks Michel Goemans and Ankur Moitra*for*discussing*the**Santa**Claus*problem with him in an early stage*of*this work. ...
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